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+{"text":"\\section{Introduction}\nMagnetic fields permeate the Universe and often play an important role in the dynamics of astrophysical processes \\citep{crutcher:12, vlemmings:13, crutcher:19}. It is difficult to directly observe magnetic fields; one typically has to use the polarization properties of the observed light \\citep[e.g.,][]{han:17}. For (sub)millimeter interferometers, such as ALMA, magnetic field detection is done mostly through dust \\citep[e.g.,][]{hull:17} and line polarization observations \\citep[e.g.,][]{vlemmings:17}. However, it has recently become increasingly clear that dust polarization does not always faithfully trace the magnetic field morphology, but instead it can be affected by processes such as self-scattering \\citep{kataoka:15, kataoka:17}. Line polarization observations are not affected by such processes, and therefore they likely trace the magnetic field structure of the observed region. However, to interpret line polarization observations, modelers have to defer to the theory of \\citet{goldreich:81}, which relies on the large velocity gradient (LVG) approximation, and therefore they cannot treat three-dimensional (3D, magnetic field) structures. \n\nIn this paper, we present POlarized Radiative Transfer Adapted to Lines (PORTAL)\\footnote{The source code of PORTAL is available on GitHub at \\url{https:\/\/github.com\/blankhaar\/PORTAL}.}, which is a 3D polarized radiative transfer code that simulates the emergence of polarization in the emission of atomic or molecular (sub)millimeter lines. PORTAL can be used in stand-alone mode or process the output of regular 3D radiative transfer codes. We are able to model the emergence of linear polarization in (sub)millimeter lines through two main approximations: (i) the strong magnetic field approximation and (ii) the anisotropic intensity approximation.\\ We show that both of them are valid in the majority of astrophysical regions. \n\nRegular radiative transfer models of astrophysical environments only take the total radiation intensity and its effect on the local isotropic populations into account \\citep{vandertak:07, brinch:10}. The local populations are determined by the balance of collisional and radiative events that both excite and de-excite the populations of the molecular and atomic species. Collisional events are isotropic and a function of the density and temperature of the environment. At the outset, for unaligned quantum states, spontaneous emission events are also isotropic, that is,~the direction of the next spontaneously emitted photon of a certain molecule is random. The probability of absorption of those randomly directed photons, however, need not be isotropic \\citep{goldreich:81}. Anisotropy in the local absorption of photons aligns the quantum states that are associated with the line-transition, which in turn leads to polarization in the emission \\citep{morris:85, landi:06}. By considering the directional dependance of the photon-escape probability in a medium with an anisotropic velocity gradient, \\citet{goldreich:81} showed that radiation emitted from such a system is partially polarized. This effect is known in the literature as the Goldreich-Kylafis (GK) effect. It is strongest for lines with optical depth around unity in regions where the collisional rates are not so high as to quench the molecular or atomic alignment. Provided the magnetic field precession rate is 10-100 times stronger than radiative and collisional rates, which we show to be the case in most astrophysical regions in Section \\ref{sec:comp_align}, the line polarization traces the magnetic field projected onto the plane of the sky with a $90^{\\circ}$ ambiguity. \n\nNumerical modeling of the GK effect has been based on the theory presented in \\citet{goldreich:82}. In such models, the perpendicular and parallel components (with respect to the projected magnetic field direction) of the radiation field are propagated through a medium with an anisotropic velocity gradient. The velocity-gradient is so strong that the large velocity gradient (LVG) approximation can be employed. The LVG escape probability is a function of the velocity-gradient and is therefore anisotropic. This leads to alignment in the molecular or atomic states associated with the transition under investigation. Because of this, the emitted radiation is partially polarized. \\citet{deguchi:84} later showed that in order to accurately model the GK effect, it is vital to perform comprehensive (polarized) excitation modeling of the molecular or atomic quantum states and also of the ones that are not associated with the transition under investigation. \\citet{cortes:05} showed that an external anisotropic radiation source, such as a nearby stellar object, can enhance the polarized emission significantly. These numerical models only considered the one-dimensional propagation of polarized radiation, and the representation of the radiation field in perpendicular and parallel components is only valid when the magnetic field direction is constant over the investigated path. Furthermore, because of its heavy reliance on the LVG approximation, numerical modeling based on \\citet{goldreich:82} can only consider the introduction of anisotropy in the escape probability through an anisotropic velocity gradient. In light of recently developed polarimetric capabilities of interferometers, such as ALMA, these types of approximations cannot be afforded anymore. Rather, one needs comprehensive modeling of the 3D radiative transfer and its anisotropy, taking both the spatial and velocity structure into account for the astrophysical region under investigation as well as the 3D structure of the magnetic field. \n\nIn this paper, we demonstrate how such modeling can be attained. By using two (main) approximations, we show that regular (nonpolarized) radiative transfer codes can be extended with polarization capabilities. In Section 2, we introduce these approximations and show their simplifying impact on the theory of line polarization. In Section 3, we show how our PORTAL code provides the option of computing the emerging polarization using the output from a regular 3D radiative transfer code, in particular LIME \\citep{brinch:10}. In Section 4, we present the capabilities of PORTAL through the simulation of the emergence of polarization in a protoplanetary disk and a collapsing sphere. We discuss our results in Section 5 and conclude in Section 6.\n\n\n\\section{Theory}\nWe describe the introduction of anisotropy in the molecular or atomic populations through an anisotropic radiation field using the formalism of \\citet{landi:06}. We make the following approximations: \n\nFirst, we assume the magnetic field precession rate is way higher than collisional and radiative rates. We call this the strong magnetic field approximation. The magnetic precession rate is in the order of s$^{-1}$\/mG for diamagnetic (i.e.,~weakly magnetizable) molecules. Typical collisional rates are on the order of $10^{-5} \\left(\\frac{n_{H_2}}{10^6 \\ \\mathrm{cm}^{-3}}\\right)\\ \\mathrm{s}^{-1}$ and radiative rates are on the order of $10^{-4} \\ \\mathrm{s}^{-1}$ for a transition at $100$ GHz with a dipole moment of $0.1$ Debye, which is shone upon isotropically by $400$ Kelvin black-body radiation. Therefore, for almost all molecules, magnetic field interactions already dominate at very weak magnetic fields ($\\mu$G). Under the assumption of a strong magnetic field, many terms in the polarized density-equations can be dropped \\citep{landi:06}. The strong magnetic field approximation is also invoked by \\citet{goldreich:81}. In Section \\ref{sec:comp_align} we discuss special cases where a dominant magnetic field cannot be assumed.\n\nSecond, we assume that only the total intensity of the radiation has an influence on the (polarized) populations of the molecular or atomic states. This is a reasonable assumption if the polarization fraction is low, which is corroborated by polarization observations of molecular emission lines. We refer to this approximation as the anisotropic intensity approximation. We discuss the validity of the anisotropic intensity approximation in more detail in Section \\ref{sec:anis_int}, where we also compare our modeling with that of \\citet{goldreich:81}, who take the influence of both the Stokes-I and -Q parameters on the alignment of the molecular states into account. \n \nThese assumptions lead to significant simplifications in the theory behind the alignment of molecular and atomic quantum states and the radiation with which they interact. They allow for the implementation of such a model as an extension to a regular line radiative transfer code. In the following, we introduce the formalism that we used to model the alignment to molecular or atomic quantum states. After this, we outline how aligned quantum states influence the propagation of polarized radiation. \n\\subsection{Polarized statistical equilibrium equations}\nThe polarizing mechanism we focus on is the anisotropic radiation field. Mathematically, anisotropy in the radiation field that affects the quantum state alignment is most easily described in terms of an irreducible tensor-element expansion. The irreducible tensor components of the radiation field, which are in direction $\\Omega$ and at the position $\\boldsymbol{r}$, are obtained as \\citep{landi:84}\n\\begin{align}\n\\mathcal{J}^K_Q (\\boldsymbol{r},\\nu , \\Omega) = \\sum_j \\mathcal{T}^K_Q(j,\\Omega) S_j(\\boldsymbol{r},\\nu,\\Omega),\n\\label{eq:irred_J}\n\\end{align}\nwhere $K$ represents the irreducible tensor rank and $Q$ is its projection, $S_j (\\nu,\\Omega)$ are the Stokes-parameters at frequency $\\nu,$ and $j$ runs over all four Stokes parameters. We define the Stokes parameters in relation to the complex electric field vector components as \n\\begin{subequations}\n\\begin{align}\nI = |E_x|^2 + |E_y|^2 , \\\\\nQ = |E_x|^2 - |E_y|^2 , \\\\ \nU = 2\\mathrm{Re}\\left[E_x E_y^* \\right], \\\\\nV = 2\\mathrm{Im}\\left[E_x E_y^* \\right], \n\\end{align} \n\\end{subequations}\nwhere $x$ and $y$ refer to the axes that are perpendicular to the propagation direction, $z$, and each other. In this work, we consistently chose the axis of $x$ along the rejection of the (local) magnetic field direction from the propagation direction. The transformation coefficients $\\mathcal{T}^K_Q(j,\\Omega)$ are defined in equation~(A6) from \\citet{landi:84}. If we only consider alignment by Stokes-I radiation and if we furthermore assume a dominant magnetic field, only the $K=0,2$ and $Q=0$ components are of interest \\citep{landi:06}. Under these conditions, the irreducible tensor components of the radiation field reduce to\n\\begin{subequations}\n\\begin{align}\n\\mathcal{J}^0_0(\\boldsymbol{r},\\nu,\\Omega) &= I(\\boldsymbol{r},\\nu,\\Omega), \\\\\n\\mathcal{J}^2_0(\\boldsymbol{r},\\nu,\\Omega) &= \\sqrt{\\frac{1}{2}}P_2 (\\mu ) I(\\boldsymbol{r},\\nu,\\Omega),\n\\end{align}\n\\label{eq:J_int}\n\\end{subequations}\nwhere $\\Omega = (\\theta,\\phi)$ is expressed in terms of the inclination and azimuth angles that are gauged with respect to the magnetic field direction. The quantity $P_2 (\\mu)$ is the second-order Legendre polynomial and $\\mu=\\cos \\theta$. The solid-angle integrated tensors at position $\\boldsymbol{r}$ are readily obtained as\n\\begin{subequations}\n\\begin{align}\nJ^0_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi}\\int_{-1}^1 d \\mu \\int_0^{2\\pi} d\\phi \\  I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu),\\phi), \\\\ \nJ^2_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi \\sqrt{2}} \\int_{-1}^1 d \\mu \\ P_2 (\\mu) \\int_0^{2\\pi} d\\phi \\  I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu),\\phi). \n\\end{align}\n\\label{eq:int_rad_tens} \n\\end{subequations}\nIn the following, we refer to the ratio $J_0^0 (\\boldsymbol{r},\\nu) \/ J_0^2 (\\boldsymbol{r},\\nu)$ as the relative alignment of the radiation field. For an isotropic radiation field ($I(\\boldsymbol{r},\\nu,\\Omega) = I(\\boldsymbol{r},\\nu)$), it should be noted that only the (isotropic) $J^0_0 (\\boldsymbol{r},\\nu)$-term survives.\n\nJust as for the radiation field, we represent the molecular or atomic quantum states as irreducible tensor elements in order to most optimally utilize their symmetry properties. Quantum states are denoted as $\\rho^K_Q (\\alpha J)$, where $K$ is the rank of the irreducible tensor element and $Q$ is its projection. The total angular momentum of the associated quantum state is $J$ and all other quantum numbers characterizing the quantum state are collected in $\\alpha$. The rank $K$ is positive and restricted to values of $\\leq 2J$. The elements $K\\geq 1$ of the population tensor relate to the alignment of the quantum state and the $K=0$ element relates to the population of the quantum state. Under the assumption of a strong magnetic field, we can neglect all but the $Q=0$ projection elements. Because of the symmetry of the radiation field, we only have to take elements into account where $K$ is even. \\citet{landi:06} presented the statistical equilibrium equations for the polarized quantum state $\\rho^K_0 (\\alpha, J)$ under the following conditions: \n\\begin{align}\n\\dot{\\rho}^K_0 (\\alpha J) &= \\sum_{\\alpha_l J_l K_l} \\rho^{K_l}_0 (\\alpha_l J_l) \\left[ [t_A]_{\\alpha J K}^{\\alpha_l J_l K_l} + \\sqrt{\\frac{[J_l]}{[J]}} \\delta_{K,K_l} [C_I^{(K)}]_{\\alpha J}^{\\alpha_l J_l} \\right] \\nonumber \\\\ \n&+  \\sum_{\\alpha_u J_u K_u} \\rho^{K_u}_0 (\\alpha_u J_u) \\left[ [t_S]_{\\alpha J K}^{\\alpha_u J_u K_u} + [t_E]_{\\alpha J K}^{\\alpha_u J_u K_u} \\right. \\nonumber \\\\ \n&+ \\left. \\sqrt{\\frac{[J_u]}{[J]}} \\delta_{K,K_u} [C_S^{(K)}]_{\\alpha J}^{\\alpha_u J_u} \\right] \\nonumber \\\\\n&-  \\sum_{K'} \\rho^{K'}_0 (\\alpha J) \\left[ [r_A]_{\\alpha J K K'} + [r_E]_{\\alpha J K K'} + [r_S]_{\\alpha J K K'} \\right. \\nonumber \\\\\n&+ \\left. \\delta_{KK'}\\left( \\sum_{\\alpha_u J_u} [C_I^{(0)}]_{\\alpha_u J_u}^{\\alpha J} + \\sum_{\\alpha_l J_l} [C_S^{(0)}]_{\\alpha_l J_l}^{\\alpha J} + D^{(K)} (\\alpha J)\\right) \\right].\n\\label{eq:stateq}\n\\end{align}\nIn Eq.~(\\ref{eq:stateq}), the rate of radiative absorption events toward the $\\rho^K_0 (\\alpha, J)$ from lower level $ \\rho^{K_l}_0 (\\alpha_l J_l)$ is given by $[t_A]_{\\alpha J K}^{\\alpha_l J_l K_l}$ \nand the collisional contribution is $[C_I^{(K)}]_{\\alpha J }^{\\alpha_l J_l }$. The rate of stimulated and spontaneous emission events toward the $\\rho^K_0 (\\alpha, J)$ from upper level $ \\rho^{K_u}_0 (\\alpha_u J_u)$ are given by $[t_S]_{\\alpha J K}^{\\alpha_u J_u K_u}$ and $[t_E]_{\\alpha J K}^{\\alpha_u J_u K_u}$,\nand the collisional contribution is $[C_S^{(K)}]_{\\alpha J}^{\\alpha_u J_u}$. The rates of absorption, stimulated emission, and spontaneous emission from the level $\\rho^K_0 (\\alpha, J)$ to all other levels is given by $[r_A]_{\\alpha J K K'}$, $[r_S]_{\\alpha J K K'}$, and $[r_E]_{\\alpha J K K'}$.\nFinally, the collisional depolarization rates are $D^{(K)}(\\alpha J)$. More detailed expressions for the radiative rates from Eq.~(\\ref{eq:stateq}) can be found in equations 7.20 from \\cite{landi:06}. By assuming a steady-state, $\\dot{\\rho}^K_0 (\\alpha J) = 0$, the statistical equilibrium equations can be solved as a linear set of equations. The solution yields the quantum state populations, including their relative alignment. \n\nWe should note that the statistical equilibrium equations of Eq.~(\\ref{eq:stateq}) are isomorphic to those presented in \\citet{deguchi:84}. While \\citet{deguchi:84} set up the statistical equilibrium equations in the standard angular momentum basis $\\ket{jm}$, where $j$ is the total angular momentum of the eigenstate and $m$ is its projection, we worked in a spherical tensor representation. We refer to \\citet{landi:06} for a detailed discussion on the relation between the two representations. We chose to work in a spherical tensor representation because of its symmetry properties. The properties of the spherical tensor expansion of both the molecular (or atomic) states and the radiation are such that truncation of higher-order $K$-terms in the $\\rho^K_0 (\\alpha J)$-expansion can be done with minimal loss of accuracy in the description of the statistical equilibrium equations for our system. Such truncation is not possible in the representation that \\citet{deguchi:84} used, and it results in a rapid and unmitigable increase in computational effort when high angular momentum states are considered. \n\\subsection{Polarized radiative transfer}\nAfter having obtained the (aligned) quantum state populations, we can evaluate their impact on the radiation propagation. Because of the strong magnetic field, (locally) only Stokes-Q radiation is produced.\nThe propagation of radiation around frequency, $\\nu_{\\alpha' J' , \\alpha J}$, associated with a transition $\\alpha' J' \\to \\alpha J$, can be described by\n\\begin{align}\n\\frac{d}{ds} \\boldsymbol{I}_{\\nu} = -\\boldsymbol{\\kappa}_{\\nu}^{\\alpha' J' , \\alpha J} \\boldsymbol{I}_{\\nu} + \\boldsymbol{\\epsilon}^{\\alpha' J' , \\alpha J},\n\\label{eq:polrad} \n\\end{align}  \nwhere $\\boldsymbol{I}_{\\nu}=[I_{\\nu},Q_{\\nu},U_{\\nu},V_{\\nu}]$ is the Stokes vector and the propagation matrix \n\\begin{align}\n\\boldsymbol{\\kappa}_{\\nu}^{\\alpha' J' , \\alpha J} = \\begin{bmatrix}\\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & \\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) & 0 & \\eta_V^{\\alpha' J' , \\alpha J} (\\nu) \\\\ \\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & 0 & 0 \\\\ 0 & 0 & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) & 0 \\\\ \\eta_V^{\\alpha' J' , \\alpha J} (\\nu) & 0 & 0 & \\eta_I^{\\alpha' J' , \\alpha J} (\\nu) \\end{bmatrix} \n\\label{eq:kappa_mat}\n\\end{align}\nis significantly simplified if one assumes a dominant magnetic field. Because we only consider diamagnetic molecules with Zeeman splitting that are far weaker than the thermal broadening, the production of Stokes-V radiation through the Zeeman effect is negligible and we set $\\eta_V^{\\alpha' J' , \\alpha J} (\\nu) \\to 0$. Thus, in PORTAL, we only consider the propagation of linearly polarized radiation. The expressions for the $\\eta$-elements of Eq.~(\\ref{eq:kappa_mat}) are \\citep{landi:84} \n\\begin{subequations}\n\\begin{align}\n\\eta_I^{\\alpha' J' , \\alpha J} (\\nu) &= \\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} B_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}}(\\nu) \\left\\{  \\left( \\mathcal{N}_{\\alpha' J'}  - \\mathcal{N}_{\\alpha J} \\frac{[J']}{[J]}\\right) \\right. \\nonumber \\\\\n&+ \\left. \\left(\\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') - \\frac{[J']}{[J]} \\mathcal{N}_{\\alpha J} w_{JJ'}^{(2)} \\sigma^2_0 (\\alpha J) \\right) \\right. \\nonumber \\\\\n& \\times \\left. \\frac{3 \\cos^2 \\theta - 1}{2\\sqrt{2}} \\right\\}, \\\\\n\\eta_Q^{\\alpha' J' , \\alpha J} (\\nu) &= -\\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} B_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\left( \\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\right. \\nonumber \\\\ \n&- \\left. \\frac{[J']}{[J]}\\mathcal{N}_{\\alpha J} w_{JJ'}^{(2)}  \\sigma^2_0 (\\alpha J) \\right) \\frac{3\\sin^2 \\theta }{2\\sqrt{2}},\n\\end{align}\n\\label{eq:eta}\n\\end{subequations}\nwhere $\\mathcal{N}_{\\alpha J} = \\mathcal{N} [J]^{1\/2} \\rho_0^0 (\\alpha J)$ is the number density of the quantum state $\\alpha J$, and $\\phi_{\\nu_{\\alpha' J' , \\alpha J}}$ denotes the normalized line-profile centered at $\\nu_{\\alpha' J' , \\alpha J}$ in frequency-space. The symbols \n\\begin{align}\nw_{J'J}^{(2)} = (-1)^{1+J+J'}\\sqrt{3[J']}\\begin{Bmatrix} 1 & 1 & 2 \\\\ J' & J' & J\\end{Bmatrix} \\nonumber\n\\end{align}\nwere introduced by \\citet{landi:84}. The quantity between curly brackets is a Wigner-6j symbol \\citep{biedenharn:81}. We use the short-hand notation\n$\n\\sigma_0^2 (\\alpha J) = \\rho_0^2 (\\alpha J) \/ \\rho_0^0 (\\alpha J)\n$ \nfor the relative alignment of the quantum state $\\alpha J$. The spontaneous emission events in the polarized radiative transfer equations of Eq.~(\\ref{eq:polrad}) are represented in the $\\boldsymbol{\\epsilon}$-vector. The spontaneous emission contribution to the Stokes-U is zero in the strong magnetic field limit we consider. The Zeeman effect for diamagnetic molecules is way smaller than the thermal broadening, so we can set $\\epsilon_V^{\\alpha' J', \\alpha J} \\to 0$. The contributions to the Stokes-I and -Q parameters are \\citep{landi:84}\n\\begin{subequations}\n\\begin{align}\n\\epsilon_I^{\\alpha' J' , \\alpha J} (\\nu) &= \\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} A_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\nonumber \\\\\n& \\times \\mathcal{N}_{\\alpha' J'} \\left\\{  1  + w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\frac{3 \\cos^2 \\theta - 1}{2\\sqrt{2}} \\right\\}, \\\\\n\\epsilon_Q^{\\alpha' J' , \\alpha J} (\\nu) &= -\\frac{h \\nu_{\\alpha' J',\\alpha J}}{4\\pi} A_{\\alpha' J' , \\alpha J} \\phi_{\\nu_{\\alpha' J' , \\alpha J}} (\\nu) \\nonumber \\\\\n&\\times\\mathcal{N}_{\\alpha' J'} w_{J'J}^{(2)} \\sigma^2_0 (\\alpha' J') \\frac{3\\sin^2 \\theta }{2\\sqrt{2}}.\n\\end{align}\n\\label{eq:eps}\n\\end{subequations}\n\\section{Methods}\n The formalism that we present in the previous section can be used to simulate the emergence of polarization in spectral lines using an (isotropic) excitation model as input. It can be directly used by assuming LTE excitation, or alternatively, the atomic and molecular excitation from any (3D) radiative transfer code can be input. We outline in the following how we used the LIME radiative transfer code \\citep{brinch:10}. \n\nLIME is a Monte Carlo 3D radiative transfer code that works with a (weighted) randomly chosen grid. A physical structure can be input, whereupon a random grid is chosen that is weighed over the molecular density and other parameters \\citep{ritzerveld:06}. After a number of Monte Carlo radiative transfer iterations, which are sped up by an accelerated lambda iteration \\citep{rybicki:91}, the simulation converges on a molecular and atomic excitation over all of the nodes in the simulation. Subsequently, this solution can be ray traced to simulate an image of the physical structure under investigation. \n\nRather than directly ray tracing the excitation solution, we used it to thoroughly map out the local anisotropy of the radiation field throughout the simulation. With the local anisotropy parameters of the radiation field, we modeled the polarized excitation of the molecular or atomic states under investigation. Having the polarized excitation mapped out throughout the simulation, we performed a polarized ray-tracing to obtain a polarized image of the physical structure under investigation. \n\nIn the following, we outline in more detail how we implemented PORTAL. In the first paragraph, we discuss setting up the polarized statistical equilibrium equations using the output of a line radiative transfer code. In order to do this, we dedicated most of our attention to the mapping of the local anisotropy of the radiation fields. In the second paragraph, we detail the polarized radiative transfer that was performed in the polarized ray-tracing. Especially for simulations with nonuniform magnetic fields, it is crucial to pay extra attention to the frame of reference of the polarized radiation and the proper way to relate different frames of reference.        \n\n In PORTAL, we used the anisotropic intensity approximation and formulated the polarized statistical equilibrium equations in terms of irreducible tensor elements. This approach differs from other efforts such as LinePol \\citep[][(submitted)]{kuiper:20}, which builds on LIME, is optimized for CO, and uses the formalism of \\citet{goldreich:82} to describe the propagation of polarized radiation and its interaction with the molecular medium. LinePol takes two polarization modes of the radiation into account and uses a polarized accelerated lambda iteration scheme to obtain the state-populations in the simulation. At minimal cost to the accuracy of our results (see Sections \\ref{sec:polsee} and \\ref{sec:anis_int}), the approximations in PORTAL speed up the simulation tremendously and lead to the possibility to treat more complex systems. PORTAL allows for complex geometries, magnetic field structures, and the treatment of molecules with extensive energy structures.\n\n\\subsection{Polarized statistical equilibrium equations}\n\\label{sec:polsee}\nThe quantum state alignment is dependent on the local anisotropy of the radiation field, so it is important to obtain a good angular sampling of the radiation field at the location of the simulation nodes. Different angular integrations of Eqs.~(\\ref{eq:int_rad_tens}) for the case of an internal source of radiation (e.g., a central stellar object) and the case of no internal radiation source were used. For the latter, the local angular integration was performed as\n\\begin{subequations}\n\\begin{align}\nJ^0_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi}\\sum_{i=1}^{N_{\\mu}} w_i^{\\mu} \\sum_{j=1}^{N_\\phi(\\mu_i)} w_j^{\\phi} I(\\boldsymbol{r},\\nu,\\mathrm{acos}( \\mu_i),\\phi_j),  \\\\ \nJ^2_0 (\\boldsymbol{r},\\nu) &= \\frac{1}{4\\pi\\sqrt{2}}  \\sum_{i=1}^{N_{\\mu}} w_i^{\\mu} P_2 (\\mu_i)  \\sum_{j=1}^{N_\\phi (\\mu_i)} w_j^{\\phi} I(\\boldsymbol{r},\\nu,\\mathrm{acos}(\\mu_i),\\phi_j) \n\\end{align}\n\\end{subequations}\nwhere $\\mu_i$ and $w_i^{\\mu}$ are the coordinates and the weights, which were taken from the $N_{\\mu}$-point Gaussian quadrature rule. The integration over $\\phi$ was performed over $N_{\\phi} (\\mu)\\propto \\sqrt{1-\\mu^2}$ equidistant points all with weight $2\\pi\/N_\\phi$. \n\nIn the case of an internal radiation source, it should be appreciated that the solid angle associated with the radiation coming from this internal source is well-defined. Therefore, the solid angle integration was divided up into rays coming from the internal radiation source; the number of rays is proportional to the solid-angle of the internal source $\\Delta \\Omega_* = \\pi(|\\boldsymbol{r}|\/R_*)^2$ and all of the other rays were distributed equally over the remaining sphere surface.\n\nThe local radiation field parameters of a node at the position $\\boldsymbol{r}$, summarized in $J^0_0 (\\boldsymbol{r},\\nu)$ and $J^2_0 (\\boldsymbol{r},\\nu)$, were obtained by ray tracing $N$ rays with direction $\\boldsymbol{k}_{\\mu,\\phi}$ to that node. The parameters $\\mu$ were chosen with respect to the magnetic field direction ($\\boldsymbol{b} \\cdot \\boldsymbol{k}_{\\mu,\\phi} = \\mu$). The angles $\\phi$ were gauged with respect to a canonical direction not parallel to the magnetic field. The choice of the canonical direction is free as the angle $\\phi$ is integrated out without weighing (see Eq.~\\ref{eq:J_int}). The ray-tracing was performed using the molecular populations that were output by LIME, while also using some of the relevant LIME-input parameters, such as (local) temperature, (local) velocity, and gridding. The ray-tracing yielded the local radiation field parameters that were subsequently used to obtain the quantum state populations and alignment.\n\nThe quantum state populations and alignment were obtained from the statistical equilibrium equations (SEE) given in Eq.~(\\ref{eq:stateq}). The SEE are a balance of the radiative and collisional transition events. The radiative transition events are dependent on local parameters for the (an)isotopic radiation field at frequencies of all of the allowed transitions and their associated Einstein coefficients. Collisional rates are dependent on the temperature-dependent collisional cross-sections and (local) number densities of the relevant collisional partners. \nThe relevant Wigner coupling symbols were calculated using the WIGXJPF package \\citep{johansson:16}. The SEE were formulated in terms of a set of linearly dependent equations and were subsequently solved via an LQ decomposition (using the LAPACK libraries, \\citet{LAPACK}) under the following physical constraint: $\\sum_i [j_i]^{1\/2} \\rho_0^0 (\\alpha_i j_i) = 1$. The solutions also included the isotropic populations that were compared to the LIME-output. We found that neglecting the quantum state alignment terms, $\\rho_0^k (\\alpha j)$ with $k>2$, introduces an error of $\\sim 1\\% $ in the state-alignment expressions, and for $k>4$ this error is already reduced to $\\sim 1\\permil$. In general, the quantum state alignment can be neglected for terms, $\\rho_0^k (\\alpha j)$ with $k>6$, with virtually no loss in precision and with great reduction of computational effort as a consequence\\footnote{For example, the dimensionality of the polarized SEE for the first 41 rotational levels of CO reduced from 861 to 151 by setting $k_{\\mathrm{max}}=6$.}.\n\n\\subsection{Polarized radiative transfer}\nThe quantum state populations and alignment obtained from the SEE were used to compute the (polarized) absorption and emission factors for each node in the simulation. The angle $\\theta$ in Eqs.~(\\ref{eq:eta}-\\ref{eq:eps}) was obtained from the local magnetic field direction and the ray-trace direction. The ray-trace direction was chosen by defining an inclination angle and azimuth angle. The polarized radiation was gauged with respect to a canonical axis, $\\boldsymbol{\\chi}_{\\mathrm{global}}$, perpendicular to the ray-tracing direction. The local and global Stokes parameters are related as \\citep{landi:06} \n\\begin{align}\n\\begin{pmatrix} Q_{\\mathrm{local}} \\\\ U_{\\mathrm{local}} \\end{pmatrix} = \\begin{pmatrix} \\cos 2\\alpha \\ & \\sin 2\\alpha \\\\ -\\sin 2\\alpha & \\cos 2\\alpha \\end{pmatrix} \\begin{pmatrix} Q_{\\mathrm{global}} \\\\ U_{\\mathrm{global}} \\end{pmatrix},\n\\label{eq:globtoloc}\n\\end{align}\nand $I_{\\mathrm{local}} = I_{\\mathrm{global}}$. In Eq.~(\\ref{eq:globtoloc}), $\\alpha$ is the angle between $\\boldsymbol{\\chi}_{\\mathrm{global}}$ and $\\boldsymbol{\\chi}_{\\mathrm{local}}$ and the local reference axis is the unit vector along the rejection of the local magnetic field direction from the ray-tracing direction. \n\nThe local Stokes-parameters were propagated using the polarized radiative transfer equations. Equations~(\\ref{eq:polrad}-\\ref{eq:kappa_mat}) show that only the Stokes-Q and -I coefficients are coupled in the polarized radiative transfer. That means that the propagation of the Stokes-U radiation is simply $U(s) = U(0) e^{-\\eta_I s}$. To evaluate the propagation of the other Stokes parameters, $\\boldsymbol{i} = [I,Q]$, the evolution operator formalism of \\citet{landi:06} was used, where the propagation is described by\n\\begin{align}\n\\boldsymbol{i}(s) = \\int_0^s ds' \\ \\boldsymbol{O}(s,s') \\boldsymbol{\\epsilon}(s') + \\boldsymbol{O}(s,0) \\boldsymbol{i}(0),\n\\end{align} \nand where\n\\[\n\\boldsymbol{O}(s,s') = e^{\\int_{s'}^s ds'' \\  \\eta_I} \\begin{bmatrix} \\cosh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) & -\\sinh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) \\\\ \n -\\sinh \\left( \\int_{s'}^s ds'' \\ \\eta_Q \\right) & \\cosh \\left( \\int_{s'}^s ds'' \\  \\eta_Q \\right) \\end{bmatrix}\n\\]\nis the evolution operator (see Chapter 8 of \\citet{landi:84}). The propagation for each crossed cell was considered, and within such a propagation, the coefficients $\\eta_I$ and $\\eta_Q$ as well as $\\epsilon_I$ and $\\epsilon_Q$ are constant. It is then straightforward to evaluate the integrals inside the evolution operator as well as the integral over the evolution operator: $\\int_0^s ds'\\ \\boldsymbol{O}(s,s')$. Having done so, the propagation of the Stokes-I and -Q within a single cell is given by \n\\begin{subequations}\n\\begin{align}\nI(s) &= o_{I} \\epsilon_I + o_Q \\epsilon_Q + \\left[ \\cosh (\\eta_Q s) I(0) - \\sinh(\\eta_Q s) Q(0) \\right]e^{-\\eta_I s},  \\\\\nQ(s) &= o_{Q} \\epsilon_I + o_I \\epsilon_Q + \\left[ \\cosh (\\eta_Q s) Q(0) - \\sinh(\\eta_Q s) I(0) \\right]e^{-\\eta_I s}, \n\\end{align} \n\\end{subequations}\nwhere\n\\begin{align}\no_I &= \\frac{\\eta_I}{\\eta_I^2 - \\eta_Q^2} \\left(1 - \\left[\\cosh(\\eta_Q s) + \\frac{\\eta_Q}{\\eta_I} \\sinh(\\eta_Q s)\\right] e^{-\\eta_I s}\\right), \\nonumber \\\\\no_Q &= -\\frac{\\eta_Q}{\\eta_I^2 - \\eta_Q^2} \\left(1 - \\left[\\cosh(\\eta_Q s) + \\frac{\\eta_I}{\\eta_Q} \\sinh(\\eta_Q s)\\right]e^{-\\eta_I s} \\right) \\nonumber \n\\end{align}\nare the factors that were obtained from integrating the elements of the evolution operator. \n\n\n\\section{Simulations}\nWe applied PORTAL to known astrophysical problems. We consider the standard problem of a spherically symmetric collapsing molecular cloud, and we investigate the emergence of polarization in molecular lines through an anisotropic radiation field in a standard protoplanetary disk system. It should be noted that neither of these problems illustrate the full 3D capabilities of PORTAL. We focus, however, on these models because of their more straightforward interpretation and we leave more complex modeling for further work.\n \n\\subsection{Collapsing spherical cloud}\nA benchmark problem in radiative transfer modeling, which furthermore allows for local anisotropy to establish itself in the radiation field, is the problem of a collapsing spherical cloud. We consider the emergence of polarization in HCO$^+$ lines. The density, velocity, and temperature distribution are taken from the \\citet{shu:77} collapse model, using the same parameters as \\citet{zadelhoff:02}. Only the ground vibrational state of HCO$^+$ is considered. We assume a uniform HCO$^+$ abundance of $10^{-9}$ and assume constant turbulent broadening of $200 \\ \\mathrm{m\/s}$. We assume that a strong radial magnetic field (origin: center of mass) permeates the cloud. \n\nFirst of all, an overview of the relevant isotropic and anisotropic interactions is instrumental to an eventual discussion of the quantum state alignment and radiation polarization characteristics. We report the cumulative radiative and collisional rates of the $J=2$ and $J=3$ level of HCO$^+$ in Figure \\ref{fig:sphere_rates}. Of the different interactions, only stimulated emission and absorption are anisotropic interactions. Using the spherical symmetry of the collapsing sphere-problem, we only plotted the rates as a function of the distance to the center. We observe that for the inner regions of the collapsing sphere, collisions become dominant as the density of this regions increases. Even though there is appreciable alignment of the radiation field, the quantum states do not align themselves because of the dominant isotropizing collisions. From about 400 AU, radiative interactions take over as the dominant interaction and the quantum states align themselves. We also give the magnetic precession rate for a magnetic field of $1$ mG and 1 $\\mathrm{\\mu G}$ and note that for a HCO$^+$-molecule in the collapsing sphere, the magnetic field can be taken to define the symmetry axis when it is $\\sim 10-100 \\times$ stronger than other interactions. From Figure \\ref{fig:sphere_rates}, we estimate this to be the case at magnetic field strengths of $\\sim 10-100 \\ \\mathrm{\\mu G}$.\n\nIn the same Figure \\ref{fig:sphere_rates}, we plotted the relative anisotropy of the radiation field and the relative alignment of the quantum states $J=2$ and $J=3$. We note that the radiation anisotropy increases, thus moving away from the collapsing-sphere center. The radiation anisotropy in the collapsing sphere is partly a result of the density structure and partly the result of the velocity structure. Both structures are spherically symmetric, but this spherical symmetry is only manifest when the center is taken as the origin. For any cell that is not located at the center of the collapsing sphere, the radiation field is therefore anisotropic. Higher anisotropy in the radiation is associated with a stronger alignment of the quantum states.\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_rates_j2.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_rates_j3.png}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{Plot of the collapsing sphere's interaction rates (collisional, absorption, and stimulated emission as well as spontaneous emission) and relative alignment (radiative and quantum state) as a function of the radius for (a) the $J=2$ level and $J=2-1$ transition and (b) the $J=3$ level and $J=3-2$ transition. The interaction rates should be read from the left axis, the relative alignment from the right axis.}\n  \\label{fig:sphere_rates}\n\\end{figure}\n\nWe report the azimuthally averaged total intensity and polarization fraction of the HCO$^+$ $J=3-2$ and $J=2-1$ transitions in Figure \\ref{fig:sphere_frac}. Indeed, we note that close to the center of the collapsing sphere, the polarization fraction is the lowest and gradually increases when moving outward. Polarization fractions are above $1 \\%$ for a radial distance greater than $600$ AU for the $J=3-2$ transition and $900$ AU for the $J=2-1$ transition. We report the associated spectra at $R=1400$ AU in Figure \\ref{fig:sphere_spec}. We observe that the linear polarization spectra roughly follow the spectral shape of the total intensity. \n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/sphere_frac.png}\n  \\caption{Total and polarized emission intensity (in Kelvin) of a collapsing sphere as a function of the radial distance.}\n  \\label{fig:sphere_frac}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_spec_21.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/sphere_spec_32.png}\n    \\caption{}\n  \\end{subfigure}\n  \\caption{Spectra of the (polarized) intensity (in Kelvin) of the (a) $J=2-1$ and (b) $J=3-2$ transitions from a collapsing sphere. The spectra are azimuthally averaged at $1400$ AU.}\n  \\label{fig:sphere_spec}\n\\end{figure}\n\nThe polarization angles are oriented in a radial fashion along the magnetic field lines. One should be aware that for a radial magnetic field, the angle between the magnetic field lines and the propagation direction toward the observer is a function of the propagation position. Accordingly, at the magic angle of $\\theta_{\\mathrm{magic}} \\approx 54.7$ or $z\/R = \\frac{1}{\\sqrt{3}}$, the propagation elements $\\eta_Q$ flip sign and some of the earlier produced polarization is negated.\n\\subsection{Protoplanetary disk}\nThe protoplanetary disk is a prime example of an anisotropic astrophysical structure. Both the anisotropy in the density and velocity structure produce a locally anisotropic radiation field. Magnetic fields in the protoplanetary disk have been conjectured through dust-polarization observations \\citep{stephens:17}, and recently, stringent limits have been put on the magnetic field strength through ALMA line circular polarization observations \\citep{vlemmings:19}. \n\nWe consider the polarization of $^{12}$CO in a general toy model of a protoplanetary disk having a number-density distribution of \n\\begin{align}\nn_{\\mathrm{H}_2}(r_c,h) = 4\\times 10^{14} \\left(\\frac{h}{\\mathrm{AU}}\\right)^{-2.25} e^{-50 \\frac{(h\/\\mathrm{AU})^{2}}{(r_c\/\\mathrm{AU})^{2.5}}} \\ \\mathrm{m}^{-3}, \n\\end{align}\nwhere $r_c$ is the radial distance and $h$ is the height. The disk is assumed to be rotating, resulting in a model velocity-field of\n\\begin{align}\n\\boldsymbol{v}(\\boldsymbol{r})= v(\\cos \\phi \\hat{\\boldsymbol{x}} - \\sin \\phi \\hat{\\boldsymbol{y}}), \n\\end{align}\nwhere\n\\[\nv = 2.11 \\times 10^4 \\left(\\frac{r_c}{\\mathrm{AU}}\\right)^{-1} \\ \\mathrm{m\/s},\n\\]\nand $\\tan \\phi = y\/x$. The temperature is given by\n\\begin{align}\nT(r_c) = 400 \\left(\\frac{r_c}{\\mathrm{AU}} \\right)^{-\\frac{1}{2}} \\ \\mathrm{K}.\n\\end{align}  \nFurthermore, we assume a constant CO abundance of $10^{-3}$ and a constant turbulent doppler broadening of $b_{\\mathrm{turb}}=200 \\ \\mathrm{m\/s}$. We only take the vibrational ground-state of $^{12}$CO into account. We neglect any line-overlap with transitions from other species. We explore the emergence of polarization in a protoplanetary disk for three types of (strong) magnetic fields: radial, toroidal, and poloidal.\n\nWe note that perhaps this toy model of the protoplanetary disk does not capture all features of the protoplanetary disk that are important in considering the polarization of thermal lines. For instance, we neglect to represent the inner midplane region by optically thick dust, so that the anisotropic radiation field resulting therefrom is not accounted for. Also, by not taking vibrationally excited levels and the transitions between different vibrational levels into account, we fail to include their significant aligning interactions (see Section \\ref{sec:comp_align}). We explore more detailed and thorough modeling of protoplanetary disk regions in future work. These results should be taken as a simplified, but generally indicative, model of the mechanisms involved in the polarization of thermal line radiation of radiation by a magnetic field in protoplanetary disk regions. \n                                 \nIt is important to map out the rates of isotropic and anisotropic interactions in order to understand the relative alignment of the molecules or atoms. Because of the cylindrical symmetry of the protoplanetary disk, we are able to analyze the interaction rates as a function of the radial distance and the height. In Figure \\ref{fig:pp_rates}, we report the cumulative radiative and collisional rates for the $J=3$ level of CO. The rates are plotted as a function of $r_c$ for different height-cross sections. We also report the magnetic precession rate of a $1\\ \\mathrm{\\mu G}$ and a $1 \\ \\mathrm{mG}$ magnetic field. It is apparent that magnetic interactions dominate other interactions and that we are justified in choosing the projection-axis along the magnetic field direction. Further, we observe a dominance of collisions over other interactions in a large region of the inner parts of the protoplanetary disk. In the disk midplane, isotropic collisions dominate the radiative interactions in the disk, but this dominance becomes weaker with the radial distance. In the outer parts of the disk, where the density drops, collisions become weaker and radiative events dominate.\n\nIn Figure \\ref{fig:pp_rates} we also plotted the relative anisotropy of the radiation field resonant with the $J=3-2$-transition and the relative alignment of the $J=3$ state. Both of these parameters are defined with respect to a toroidal magnetic field configuration. The radiation anisotropy is strongest in the outer parts of the disk and weakest in the bulk of the disk. The same dependence is seen for the relative alignment of the quantum states. The relative anisotropy of the radiation is almost constant as a function of the radial distance at a height of $1$ AU. This is because the disk is optically thick in the midplane. The local angular radiation profile is not isotropic because of the temperature gradient. Due to dominant collisions, the quantum state alignment in the midplane is not large enough to significantly polarize radiation that is coming through. \n\n\\begin{figure}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/pp_rates_1.png}\n    \\caption{}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/pp_rates_40.png}\n    \\caption{}\n  \\end{subfigure}\n\\caption{Plot of the protoplanetary disk's interaction rates (collisional, absorption, and stimulated emission as well as spontaneous emission) and relative alignment (radiative and quantum state, with respect to a toroidal magnetic field) as a function of the radial distance for (a) $1$ AU height and (b) $40$ AU height. The interaction rates should be read from the left axis, the relative alignment from the right axis.}\n  \\label{fig:pp_rates}\n\\end{figure}\n\n\\begin{figure*}[h!]\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/rad_vec2.png}\n    \\caption{}\n    \\label{fig:pp_contour_rad}\n  \\end{subfigure}\n  ~\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/tor_vec2.png}\n    \\caption{}\n    \\label{fig:pp_contour_tor}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/rad_angle.png}\n    \\caption{}\n    \\label{fig:pp_angle_rad}\n  \\end{subfigure}\n  ~\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{.\/figures\/tor_angle.png}\n    \\caption{}\n    \\label{fig:pp_angle_tor}\n  \\end{subfigure}\n  \\caption{Contour plots of (the logarithm of) the total intensity (in Kelvin) of a protoplanetary disk. The disk is viewed face on [(a) and (b)] and at an inclination of $45^o$ [(c) and (d)]. We overlayed the intensity plot with polarization vectors from PORTAL simulations that come from a radial magnetic field (a,c) and a toroidal magnetic field (b,d). Polarization vector lengths scale with the polarization fraction.}\n  \\label{fig:pp_contour}\n\\end{figure*}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/pp_frac.png}\n  \\caption{(Polarized) emission intensity (in Kelvin) of a protoplanetary disk as a function of the radial distance. The polarized intensity is plotted for three different magnetic field configurations and the disk is seen face on.}\n  \\label{fig:pp_frac}\n\\end{figure}\n\nWe analyzed the emergence of polarization through two different magnetic field configurations: toroidal and radial. In Figure \\ref{fig:pp_contour_rad} we report the contour map of the $J=3-2$ CO-transition at $345.8$ GHz of the total intensity (in Kelvins) overlayed with polarization vectors resulting from the polarized emission of CO aligned with a radial magnetic field. The polarization vectors are scaled with respect to the polarization fraction and are parallel to the radial configuration of the magnetic field. Figure \\ref{fig:pp_contour_tor} gives the polarization map coming from a toroidal magnetic field. We note that the polarization fraction for the face-on view of the protoplanetary disk is cylindrically symmetric.\n\nIt is striking that the polarization vector maps viewed face on, for both the toroidal and radial magnetic field, have the same configurations. This similarity can be traced back to the anisotropy introduced in the molecular states via the anisotropic radiation field, $J^2_0 (\\boldsymbol{r})$ (Eq.~\\ref{eq:int_rad_tens}). When performing the integration to acquire $J^2_0 (\\boldsymbol{r})$, the $\\mu$-angle is gauged with respect to the magnetic field direction. The different gauges with respect to the toroidal and radial magnetic field configurations lead to the $J^2_0 (\\boldsymbol{r})$, which is associated with the toroidal magnetic field, to be negative, while the $J^2_0(\\boldsymbol{r})$ of the radial magnetic field is positive. Thus, in the region where polarization is produced, where furthermore the angle between propagation and the magnetic field $\\theta_{\\mathrm{prop}} > \\theta_{\\mathrm{magic}}$ for both magnetic field configurations, this gives rise to perpendicular and parallel orientations of the polarization vectors with respect to the toroidal and radial magnetic fields, that is,~polarization vectors that are identically oriented. Only when we view the disk at a significant inclination are we able to discern the orientation of the magnetic field from its polarization vectors, which can be seen in Figures \\ref{fig:pp_angle_tor} and \\ref{fig:pp_angle_rad}. \n\nThe polarization maps of a protoplanetary disk viewed at a $45^o$ inclination show large polarization fractions for the poloidal and toroidal magnetic field configurations. Lower but still significant polarization fractions are seen to emerge from the radial magnetic field configuration. The highest polarization fractions occur at the edges of the protoplanetary disk. In the disk midplane, almost no polarization arises. This effect can be ascribed to the high optical depth from this region; it should, however, also be noted that our method underestimates the polarization fraction coming from optically thick regions (see Section \\ref{sec:anis_int}). \n\nFor the face-on view of a protoplanetary disk that is permeated by a poloidal magnetic field, no significant polarization emerges even though the quantum states are aligned. This is because for a large part of the disk, the magnetic field is almost aligned along the propagation direction. When this is the case, the propagation coefficients are $\\eta_Q \\to 0$, and no polarization is produced. When the disk is viewed at a significant inclination, the poloidal magnetic field produces a large polarization fraction. \n\nFigure \\ref{fig:pp_frac} is a plot of the azimuthally averaged polarization fraction as a function of the radial distance. Near the center of the proto-planetary disk, the polarization fraction is low and increases as one moves outward. The maximum polarization fraction of the protoplanetary disk viewed face on is  $\\sim 0.5 \\%$, but polarization fractions up to $\\sim 9\\%$ are observed when the disk is viewed at an inclination of $45^o$. We analyze the azimuthally averaged ($r_c = 50$ AU) spectrum of the total (polarized) intensity in Figure \\ref{fig:pp_spec}. The polarization roughly follows the spectral shape of the total intensity.\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/pp_spec.png}\n  \\caption{Spectrum of the (polarized) intensity (in Kelvin) of the $J=3-2$ transition from a protoplanetary disk permeated by a toroidal magnetic field. The spectrum is azimuthally averaged at $60$ AU and the disk is seen face on.}\n  \\label{fig:pp_spec}\n\\end{figure}\nIt is a general trend that high-frequency transitions have a larger tendency to emit polarized radiation. This is because the radiative rates scale with the frequency. Radiative interactions of high-frequency transitions therefore tend to dominate over collisional interactions. At the same time, the transition optical depth falls (generally) with the transition frequency; for transitions that are too optically thin, radiation intensity is too low to align the quantum states. \n\n\\section{Discussion}\nThe anisotropic intensity approximation and the strong magnetic field approximation are central to the quality of the method we employed in PORTAL. We discuss these two approximations in the following two subsections. We discuss general remarks about the simulations of astrophysical regions using PORTAL in Section \\ref{sec:general_remarks}. \n\\subsection{The anisotropic intensity approximation}\n\\label{sec:anis_int}\nOur method heavily relies on the approximation that it is only the anisotropy in the total intensity that contributes to the alignment of the molecular or atomic states under investigation. We call this approximation the anisotropic intensity approximation. We were able to directly compare the anisotropic intensity approximation to the LVG problem of \\citet{goldreich:81}. \\citet{goldreich:81} accounted for the influence of the anisotropy of both the Stokes I and Stokes Q on the quantum state alignment. In the GK approach, the Stokes U component of the radiation field is neglected because the LVG method can only treat a constant magnetic field. The comparison is summarized in Figure \\ref{fig:GK_compare}.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{.\/figures\/compare_GK.png}\n  \\caption{Comparison of the polarization fraction computed through the GK method (solid line) and the radiation anisotropy method we employ in this paper (dotted line). For more details on the simulation parameters, see \\citet{goldreich:81}. We consider a $J=1-0$ transition at $100$ GHz, with a strong magnetic field along the $\\hat{z}$-axis and a velocity gradient of $10^{-9}$ s$^{-1}$ in the $xy$-plane. We consider a temperature of $T=10$ K. Three ratios for the collision-radiative rates are considered and denoted inside the figure. The polarization fraction was computed for a ray traveling along the $\\hat{x}$-axis.}\n  \\label{fig:GK_compare}\n\\end{figure}\n\nWe note that below polarization fractions of $2\\%$, our method agrees with the GK effect for any optical depth. Furthermore, for low optical depth, $\\tau < 0.3$, our method reproduces the GK effect very well regardless of the polarization fraction. It is only for very high degrees of polarization and large optical depths that the polarization fraction obtained through the anisotropic intensity approximation starts to deviate from the GK polarization fraction. For strongly polarized lines ($p_L > 6\\%$), the polarization fraction can be underestimated by up to a factor $1.5$ for $\\tau > 1$ and this underestimation is sustained with increasing $\\tau$. We note that the polarization angle is identical for both methods.\n\nThe anisotropic intensity approximation loses its quality through the following: (i) the fact that a significant part of the radiation is polarized, which has an impact on the irreducible tensor representation of the radiation field (see Eq.~\\ref{eq:irred_J}), and (ii) that this simplification subsequently impacts the source function, resulting in a magnification of the error. The latter error is particularly manifest at high optical depths, and it is also a consequence of the local approximation of an LVG-like problem. We expect this error to be ameliorated when the local approximation is abandoned as in PORTAL. \n\nThe polarization of (sub)millimeter lines through the GK effect has been observed in a number of sources. For most line observations, the observed polarization fraction is lower than $2\\%$ \\citep{lai:03}. This can be taken as a direct indicator of the quality of the anisotropic intensity approximation. There is a fraction of emission lines for which high polarization fractions are observed; the most strongly polarized emission lines go up to $13 \\%$ \\citep{vlemmings:12, cortes:05}. The large polarization fractions are most probably due to large sources of external radiation in the vicinity. \n\nOne avenue to remedy the anisotropic intensity approximation is to iteratively perform the inward ray-tracing steps (see Section \\ref{sec:polsee}) for all radiative polarization modes and perform the irreducible tensor integration as Eq.~(\\ref{eq:irred_J}). After each iteration, the alignment of the quantum states for each cell is recomputed until convergence is attained. We plan to implement such a scheme in a later version of PORTAL, although this will significantly increase the calculation time.\n\\subsection{The strong magnetic field approximation}\n\\label{sec:comp_align}\nThe symmetry axis of the molecular and atomic states determines the (projected) direction of polarization. In our models, it is assumed that the symmetry axis is along the local magnetic field direction. This requires the magnetic precession rate to be $10$-$100$ times stronger than other directional interaction rates. If an alternative directional interaction is about as strong or stronger than the magnetic precession rate, then the symmetry axis of the quantum states is rotated. \n\nThe magnetic precession rate for a nonparamagnetic molecule is given by \n\\begin{align}\ng\\Omega = 4.8 g_{\\mathrm{mol}} \\left(\\frac{B}{\\mathrm{mG}} \\right) \\ \\mathrm{s}^{-1}, \n\\end{align} \nwhere $g_{\\mathrm{mol}}$ is the molecular g-factor: A dimensionless factor that determines the coupling of the molecule to the magnetic field. For linear molecules, $g_{\\mathrm{mol}}$ is the same for all rotational levels. The molecular g-factors of CO and HCO$^+$ that we consider in this work are $g^{\\mathrm{CO}} = -0.269$ \\citep{flygare:71} and $g^{\\mathrm{HCO}^+}=0.006$.\\footnote{\nWe computed the g-factor of HCO$^+$ using quantum chemical techniques since no experimental data are available. The quantum chemical calculations were performed at the CCSD(T) level of theory, using aug-cc-pVTZ basis sets, with the CFOUR program package \\citep{CFOUR}. We used a linear geometry of $r_{\\mathrm{CO}}=1.112$ \\AA and $r_{\\mathrm{CH}}=1.095$ \\AA. We note that the molecular g-factor of HCO$^+$ is anomalously low. Indeed, the only polarimetric observation of HCO$^+$ yielded no detection \\citep{glenn:97}. This could be an effect of weak Zeeman precession. However, one should not forget that HCO$^+$ has in fact a hyperfine structure, where each hyperfine-transition has its own g-factor \\citep[see, for instance,][]{lankhaar:18} that only averages to the rotational g-factor if all hyperfine-transitions have line-strengths proportional to the hyperfine-resolved Einstein A-coefficients.\n}\n\nWe compare the magnetic precession rate ($1$ mG and $1$ $\\mathrm{\\mu G}$) to the cumulative rate of stimulated emission in Figures \\ref{fig:sphere_rates} and \\ref{fig:pp_rates}. For the problems we considered, the magnetic precession rate is dominant over all other interactions and it is justified to assume that the quantum state symmetry axis is along the magnetic field direction.\n\nEarlier, we saw that HCO$^+$ had an exceptionally low magnetic moment. Conversely, the dipole moment of HCO$^+$ is very large. Thus radiative interactions for such a molecule are very strong, and therefore also a strong magnetic field is required to justify the dominant magnetic field approximation. Indeed, for a large region of the collapsing sphere, a $1 \\ \\mathrm{\\mu G}$ magnetic field would not determine the HCO$^+$ symmetry axis. We stress that for molecules that have strong radiative interactions, one should be extra vigilant and check the relevant interaction rates to verify that the magnetic field truly defines the symmetry axis of the quantum states and thus if the polarization vectors do indeed trace the magnetic field structure. \n\nIt is conceivable that a strong external radiation field that has a large angular size, such as a large stellar object, determines the quantum state symmetry axis. The directional rate of interaction of a general lower quantum state, $1$, by an external black-body radiation source at the solid angle $\\Delta \\Omega_*$ and with the temperature $T_*$ is \\citep{nedoluha:92,morris:85} \n\\begin{align}\nR_{12} = \\frac{g_2}{g_1} A_{21} \\left[e^{h\\nu_{21} \/ k_B T_*} - 1 \\right]^{-1} \\Delta \\Omega_*,\n\\end{align}\nwhere $g_i$ is the degeneracy of level $i$ and $A_{21}$ and $\\nu_{21}$ are the Einstein coefficient and frequency of the transition from upper level $2$ to lower level $1$. It is apparent from this expression that (sub)millimeter lines have relatively low interaction rates. Rather, vibrational transitions in the IR region have associated directional interaction rates that are far greater and are more likely to compete with magnetic interactions to determine the symmetry axis of the quantum states. For instance, the interaction rate of the $(v,J)$, $(0,0)\\to (1,1)$ transition of CO is $\\sim 7.8 \\ \\mathrm{s}^{-1}$ when it is excited by a $2000$ Kelvin black-body radiation source at $\\Delta \\Omega_* = 1$ sr. The rate drops quadratically with the distance to the external radiation source and it is not corrected for absorption. We implemented a module in PORTAL that can incorporate the interactions resulting from a bright external source of radiation through vibrational transitions. This is particularly important when investigating the circumstellar envelopes of evolved stars \\citep{morris:85, ramos:05}\n\nThe strong magnetic field approximation should be abandoned when multiple directional interactions have similar interaction rates. In that case, one must comprehensively model all anisotropies affecting the quantum state alignment. This can be done at the expense of a computational effort as it increases the dimensionality of the problem greatly. For example, in treating the first $41$ rotational levels of a linear rotor and by setting $k_{\\mathrm{max}}=6$, the dimensionality of the SEE would increase from $151$ to $1086$, provided that we neglect orientation elements of uneven $k$. The general theory of setting up the complete SEE can be found in Chapter 7 of \\citet{landi:06}. \n\\subsection{General remarks}\n\\label{sec:general_remarks}\n\\subsubsection{(Sub)millimeter line polarization in astrophysical regions}\nIt is clear from our calculations that the only requirement for the emergence of polarized emission is a source that has some form of anisotropy. This anisotropy may come from the velocity field, which has already been explored by \\citet{goldreich:81}, but it is not necessarily limited to this. To present the capabilities of PORTAL, we computed the emergence of polarized radiation in a protoplanetary disk and a collapsing sphere. In the protoplanetary disk, anisotropy mostly comes from the density structure. For the collapsing sphere, anisotropy comes from both the velocity-field and the density structure.\n\nFurthermore, we confirm the earlier observation of \\citet{goldreich:81}, which is that namely around optical depths of unity, the polarization of line emission is the strongest. The physical reason behind this is that for sources with some sort of anisotropy, around $\\tau \\sim 1,$ this anisotropy is most manifest in the local radiation field. Subsequently leading to the highest polarization degrees. \n\n\\subsubsection{Sampling} \nThe sampling of the space that we used is identical to the sampling used by LIME in which a random sampling, weighed by the density-structure, of the space is performed and neighboring cells are found through a Voronoi tessellation. We found that the extensive angular sampling that we performed to compute the local anisotropic radiation field generally requires a higher sampling of the space than would be necessary if one is generating a nonpolarized image. We found that for insufficient sampling of the space, strong local variation in the polarization fractions manifest themselves even though similar variations would not be visible in the total intensity. Also, local $90^o$ flips of the polarization vectors can be a product of sampling of the surrounding space that is too sparse. For a source with symmetry in both the magnetic field and the radiative transfer structure, it can be prudent to use symmetrical averages in the case of a sampling that is too sparse.\n\n\\subsubsection{Collisions} \nIn order for appreciable polarization in the emission from astrophysical regions to be produced, one requires the rate of (isotropic) collisions to be relatively low. When collisions occur more than $100$ times as frequent as the aligning absorption and stimulated emission events, no observable polarization is produced. Polarization is therefore not produced in regions of high number density and temperature. In general, regions that are in local thermal equilibrium show no appreciable polarization in their emission. \n\nIn the astrophysical problems that we analyzed, we represented collisions only by their rank-0 elements, that is, we assumed all magnetic substates to be equally pumped. At the same time, we assumed no depolarization through elastic collisions. The systematic errors of both assumptions are opposite. Such an approximation for the alignment characteristics of collisional rates is a common assumption in the modeling of alignment of quantum states \\citep{landi:06}. Indeed, collisional rates resolved at the level of magnetic substates are not readily available, even though it is possible to compute these using modern quantum-dynamical methods \\citep{alexander:79, faure:12, landi:06}. \n\n\n\\subsubsection{External radiation} \nWe found that an external source of directional radiation enhances the polarization appreciably. Similar conclusions have also been drawn in maser polarization theory \\citep{lankhaar:19} and also for the GK effect \\citep{deguchi:84, cortes:05}. In particular, \\citet{cortes:05} found that they could explain the $90^o$-flip in polarization angle between the CO $J=1-0$ and the $J=2-1$ transitions through the anisotropic radiation coming from an external source. We confirm that this is one possible explanation, but we stress that there are other avenues to attain such a polarization effect. According to our theory, this $90^o$-flip is most generally explained by the $\\eta_Q^{1-0}$ and the $\\eta_Q^{2-1}$ elements being of opposite signs. This does not necessarily require an external radiation source. \n\nIt should be emphasized that polarization enhancement through external radiation is most manifest when hot objects irradiate high-frequency transitions, such as vibrational lines. It is also the case for such transitions that are most likely to compromise the strong magnetic field approximation (see Section \\ref{sec:comp_align}). In this work, we have abstained from including higher vibrational states when computing the polarization maps, but we will further explore this when we use PORTAL in conjunction with more detailed models of astrophysical regions and the involved radiative processes. \n\n\\subsubsection{Alternative routes to polarization} \nDust emission is often observed to be partially polarized. This has been seen in protoplanetary disks \\citep{hull:17}, in circumstellar envelopes of evolved stars \\citep{vlemmings:17}, and molecular clouds \\citep{soler:13}. Polarized emission from dust follows from its alignment. Dust can get aligned to the magnetic field through the process of radiative torque alignment \\citep{draine:97}, but alignment to a strong external source of radiation \\citep{lazarian:07} or through self-scattering \\citep{kataoka:15} is also possible. \n\nThe dust polarization is indicative of the alignment and therefore does not always trace the (projected) magnetic field direction. Polarization fractions are observed to be up to a few percent. In PORTAL, we neglected the contribution of the dust polarization to the molecular state alignment because we used the anisotropic intensity approximation. In the ray-tracing step, we implemented the dust polarization module outlined in \\citet{padovani:12} and added it to the regular line polarization ray-tracing. We have found in the simulations we present here that the contribution of the dust polarization around the line-frequency is negligible because the line-opacity is some orders of magnitude greater than the dust opacity. This means that for strong enough magnetic fields (see Section \\ref{sec:comp_align}), line polarization faithfully traces the (projected) magnetic field direction with 90$^o$ ambiguity.\n\nRecently, it has been proposed that through forward scattering of radiation by a collective of molecules, a phase difference can be induced to the parallel and perpendicularly polarized components of the radiation field \\citep{houde:13}. The phase difference subsequently leads to a conversion of Stokes-U to Stokes-V radiation. This process, called anisotropic resonant scattering, would lead to the production of circular polarization at the cost of linear polarization, and it also changes the polarization angle. Observational evidence for this phenomenon is accruing \\citep{hezareh:13, chamma:18}. Anisotropic resonant scattering is typically thought to occur in a foreground cloud, between the observer and the source of polarized line emission \\citep{houde:13}, but it could also be a feature of the radiative transfer inside the source. A better estimate of the relative strength of anisotropic resonant scattering has to be developed before we can evaluate the importance of this effect on the emergence of linear polarization in thermal line emission. \n\n\\subsubsection{Ground state alignment.} \n\\citet{yan:06} showed that polarization can emerge in atomic (hyper)fine-structure lines through (i) a strong magnetic field that defines the symmetry axis and (ii) an external UV radiation field that induces directional transitions, aligning the quantum states. If the pumping rate is much lower than the spontaneous decay rates of the excited states, only the ground state of the atomic system is aligned. Collisions and stimulated emission events are neglected in the formalism of ground state alignment (GSA). Through neglecting collisions and stimulated emission events and adapting an idealized geometry, \\citet{yan:06} are able to formulate semianalytical expressions for the polarization fractions emerging from atomic lines. GSA has been proposed as a polarizing mechanism for atomic lines in the ISM \\citet{zhang:18}.\n\nPORTAL builds on the same theory as GSA, but it explicitly incorporates the effect of collisions and stimulated emission events. Furthermore, instead of assuming that a radiation field only comes from an external source, PORTAL maps out the full 3D radiation field structure of the medium in which the investigated species is embedded. In this work, we focus on the polarized radiative transfer of (sub)millimeter molecular and atomic lines because its radiative transfer does not involve any scattering \\citep{brinch:10}. We plan to extend our model to also incorporate the emergence of polarization in atomic fine-structure lines, where we will pay special attention to scattering in the radiative transfer of these systems.\n\n\\section{Conclusions}\nWe present PORTAL, a 3D polarized radiative transfer program that is adapted to lines. The program uses the strong magnetic field approximation and the anisotropic intensity approximation, both of which we show to hold for the majority of relevant astrophysical problems. PORTAL can be used in stand-alone mode using an LTE estimate of the molecular or atomic excitation. Alternatively, the output of existing 3D radiative transfer programs can be input in PORTAL. \n\nTo outline PORTAL's capabilities, we computed the polarization maps of a collapsing sphere and a simple protoplanetary disk model. The polarization spectrum of a collapsing sphere shows polarization in its spectral lines up to $2\\%$ with the associated polarization vectors aligned with the projected magnetic field direction. The protoplanetary disk when viewed face on shows polarization fractions up to $\\sim 0.5 \\%$, but the polarization fraction rises to $\\sim 9\\%$ at significant inclinations. The polarization vectors resulting from a radial and toroidal magnetic field configuration are identical for a face-on view of the protoplanetary disk, and they can only be distinguished when viewed at a significant inclination. In forthcoming papers, we plan to use PORTAL to analyze the emergence of polarization in spectral lines in more detailed models of protoplanetary disks, to a molecular outflow, and to the circumstellar envelopes of AGB stars.\n\n\\begin{acknowledgements} Support for this work was provided by the Swedish Research Council (VR). Simulations were performed on resources at the Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC). Ko-Yun Huang and Athol Kemball are acknowledged for sharing the results of their GK code. The authors thank Luis Velilla Prieto for helpful comments on a first draft of the manuscript. We thank the referee (Martin Houde) for comments that improved the paper.\n \\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{When a star cluster meets a cloud}\n\\label{sec:1}\nWe study the encounters between stars clusters and giant molecular\nclouds (GMCs) \\cite{gieles06b}. The effect of these encounters has\npreviously been studied analytically for two cases: 1) head-on\nencounters, for which the cluster moves through the centre of the GMC\n\\cite{1987gady.book.....B} and 2) distant encounters, where the\nencounter distance $p>3\\mbox{$R_{\\rm n}$}$, with $p$ the encounter parameter and \\mbox{$R_{\\rm n}$}\\ the\nradius of the GMC \\cite{1958ApJ...127...17S}. We introduce an\nexpression for the energy gain of the cluster due to GMC encounters\nvalid for all values of $p$ and \\mbox{$R_{\\rm n}$}\\ of the form\n\n\\begin{equation}\n\\Delta E  \\simeq \\frac{4.4\\,\\mbox{$r^2_{\\rm h}$}}{\\left(p^2+\\sqrt{\\mbox{$r_{\\rm h}$}\\,\\mbox{$R_{\\rm n}$}^{3}}\\right)^2}\\,\\left(\\frac{G\\mbox{$M_{\\rm n}$}}{\\mbox{$V_{\\rm max}$}}\\right)^2\\,M_c.\n\\label{eq:1}\n\\end{equation}\nHere \\mbox{$V_{\\rm max}$}\\ is the maximum relative velocity, $\\mbox{$M_{\\rm n}$}$ is the mass of the\nGMC, $G$ is the gravitational constant and \\mbox{$r_{\\rm h}$}\\ and \\mbox{$M_{\\rm c}$}\\ are the\nhalf-mass radius and mass of the cluster, respectively. We perform\n$N$-body simulations of encounters with different $p$ and compare the\nresulting ${\\rm \\Delta} E$ of the cluster to\nEq.~\\ref{eq:1}. Fig.~\\ref{fig:1} shows the very good agreement between\nsimulations and predictions of Eq.~1.  Snapshots of one simulation are\nshown in Fig.~2.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height=3.8cm]{de_p.ps}\n\\caption{\\mbox{${\\rm \\Delta} E\/|E_0|$}\\ of a cluster as a function of $p$. The $N$-body results\n    are shown with diamonds. The result of \\cite{1987gady.book.....B}\n    and \\cite{1958ApJ...127...17S} for head-on and distant encounters\n    are shown as a filled circle and as a dashed line,\n    respectively. Eq.~\\ref{eq:1} is shown as a full line.}\n\\label{fig:1}      \n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[height=2.35cm]{snap_distant_5frames.ps}\n\\caption{Simulation of a close encounter between a GMC (grey shades)\nand a star cluster. The snapshots are viewed in the centre-of-mass\nframe of the cluster. \\mbox{$a_{\\rm c}$}\\ is the Plummer radius of the cluster.}\n\\label{fig:2}      \n\\end{figure*}\n\\vspace{-0.8cm}\n\\section{The cluster disruption time}\nFrom the simulations we find that the fractional mass loss ($\\mbox{${\\rm \\Delta} M\/|M_0|$}$) is only\n25\\% of \\mbox{${\\rm \\Delta} E\/|E_0|$}. This is because stars escape with velocities much higher\nthan the escape velocity. Defining the cluster disruption time as\n$\\mbox{$t_{\\rm dis}$}=\\mbox{$M_{\\rm c}$}\/\\dot{\\mbox{$M_{\\rm c}$}}$, we find a cluster disruption time of the form\n\n\\begin{equation}\n\\mbox{$t_{\\rm dis}$} = 2.0\\,S\\left(\\mbox{$M_{\\rm c}$}\/10^4\\,\\mbox{$M_{\\odot}$}\\right)^{\\gamma}{\\mbox{Gyr}},\n\\label{eq:2}\n\\end{equation}\nwith $S\\equiv1$ for the solar neighbourhood and scales with the global\n GMC density ($\\mbox{$\\rho_{\\rm n}$}$) as $S\\propto\\mbox{$\\rho_{\\rm n}$}^{-1}$. The index $\\gamma$ is\n defined as $\\gamma=1-3\\lambda$, with $\\lambda$ the index that relates\n the cluster half-mass radius to its mass ($\\mbox{$r_{\\rm h}$} \\propto\n \\mbox{$M_{\\rm c}$}^{\\lambda}$).  The observed shallow relation between cluster\n radius and mass (e.g. $\\lambda\\simeq0.1$), makes the index\n ($\\gamma=0.7$) similar to the index found both from observations\n \\cite{2005A&A...441..117L} and from simulations of clusters\n dissolving in tidal fields ($\\gamma\\simeq0.62$). The constant of 2.0\n Gyr, which is the disruption time of a $10^4\\,\\mbox{$M_{\\odot}$}$ cluster in the\n solar neighbourhood, is about a factor of 3.5 shorter than found from\n earlier simulations of clusters dissolving under the combined effect\n of the galactic tidal field and stellar evolution. It is only slightly\n higher than the observationally determined value of 1.3 Gyr\n \\cite{2005A&A...441..117L}, suggesting that the combined effect of\n tidal field and encounters with GMCs can explain the lack of old open\n clusters in the solar neighbourhood \\cite{1958RA......5..507O}. GMC\n encounters can also explain the (very) short disruption time that was\n observed for star clusters in the central region of M51\n \\cite{2005A&A...441..949G}, since there $\\mbox{$\\rho_{\\rm n}$}$ is an order of\n magnitude higher than in the solar neighbourhood.\n\n\t    \n\\vspace{-0.3cm}\n\\input{referenc}\n\n\n\\printindex\n\\end{document}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The Ever-Increasing Threat of Cyber Crimes}\nIn recent years, the risk of cyber crimes has emerged as a top global concern. \nThis is made evident by the world economic forum's annual global risk report \\citep{wef2020}, which regularly places cyber attacks and theft of data in its ``Top 5 global risks in terms of likelihood''.\nOver the years, the frequency and severity of cyber attacks have increased significantly globally, and they are projected to continue to increase in the future.\nRecently, Cybersecurity Ventures estimated the cost of cyber crimes to rise to 10.5 trillion USD annually by 2025 \\citep{morgan2020cybercrimecost}, up from a world economic forum estimate of 3 trillion USD for 2015.  \nFurthermore, see a detailed descriptive analysis of cyber crimes in the business sector over the last 15 years in \\citep{shevchenko2022nature}.\n\nCyber crime can be initiated by various actors, including malicious actors within institutions and external adversaries such as rogue nation states, hackers, and cyber terrorists.\nThese attacks come in a variety of forms, ranging from denial-of-service (DoS) attacks \\citep{gupta2017taxonomy}, malware \\citep{tailor2017comprehensive}, ransomware \\citep{tailor2017comprehensive}, blackmail \\citep{rid2012cyber}, extortion \\citep{young1996cryptovirology}, and more \\citep{craigen2014defining, husak2018survey}.\nSuch criminal activities are being perpetrated on a massive scale, hitting individuals as well as organisations. A large array of organisations worldwide are targeted by cyber attacks, including government agencies, universities, financial sectors, and private corporations. \nIn particular, these attacks can affect important infrastructure units that play a key role in security and safety, such as emergency services and health care.\nSuch attacks have caused breaches and significant damages to those organisations, and often adversely affect downstream users of the compromised organisations and services.\nThe damages incurred include losses attributed to outcomes such as business interruption, loss of data, reduced reputation and trust of the organisation, legal liabilities, intellectual property theft, and potential for loss of life. \n\nThe seriousness of cyber attacks has been reflected in the {U.S.} President's executive order on \\textit{Strengthening the Cybersecurity of Federal Networks and Critical Infrastructure}, which calls for a cybersecurity framework that can ``support the cybersecurity risk management efforts of the owners and operators of the Nation's critical infrastructure\".\nCyber risk from a financial and insurance perspective has also been developed under international banking and insurance regulations. \nThe Basel~III banking Accords cover cyber risk as a key component of Operational Risk capital modelling and adequacy, and the Solvency~II insurance regulations discuss the significance of an emerging cyber insurance threat that affects insurers as well as reinsurers.\nFor example, see an overview of cyber risk from a financial and insurance perspective in \\cite{peters2018understanding, malavasi2022cyber}.\n\n \n\\subsection{Lack of Product Standardisation in the Cyber Risk Insurance Market} \nAlthough many security solutions have been developed and implemented in order to detect and prevent cyber attacks, achieving a complete security protection is not feasible \\citep{lu2018managing}.\nTo address this problem, there is an increasing demand to develop the market for cyber risk insurance and to understand the structuring of insurance products that will facilitate risk transfer strategies in the context of cyber risk; see discussions in \\cite{peters2018understanding, peters2017statistical, marotta2017cyber, bohme2010modeling} and the references therein. \n\n\nAs one can see from surveys such as \\cite{marotta2017cyber} and \\cite{shetty2010competitive}, the scope of such a marketplace is still very much in its infancy, despite the fact that financial institutions rank cyber losses in their top three loss events systematically when reporting Operational Risk loss events under Basel II\/III to national regulators. \nThe reason that the cyber risk insurance market has yet to emerge with standardised products is largely due to differences in opinion as how best to mitigate and reserve against these loss events. \nFrom a technology perspective, it is common to attempt to mitigate such events in contrast to insurance or capital reserving; see discussions in \\cite{bandyopadhyay2009managers}. \nFrom a financial risk perspective, practitioners often opt for Tier I capital reserving and avoid insurance products, as the capital reduction from Operational Risk insurance is capped under Basel regulations with a haircut of 20\\%; see discussions in \\cite{peters2011impact}. This disincentivises them to purchase insurance products that have excessive premiums. \nFrom the insurance industry's perspective, there is a lack of market standardisation on insurance contract specifications that would avoid excessive premiums to be charged when the bespoke insurance products are designed. \n\n\n \n\\subsection{Our Contributions}\n\nThe three aforementioned perspectives are beginning to change and we believe that it is a suitable time to revisit the perennial question of how best to set up an insurance marketplace for cyber loss events. \nIn this paper, we introduce for the first time a Bonus-Malus framework for cyber risk insurance that provides IT-specific incentive mechanisms for encouraging sound IT governance and technology developments. \nSpecifically, we explore a class of Bonus-Malus systems in which an insured enjoys a discount in the cost of risk transfer as a result of their upfront expenses in risk reduction in the form of self-mitigation measures; and like-wise an insurer benefits from encouraging risk reduction in their risk pools to provide competitive pricing of insurance premium. \nUnder this framework, we also demonstrate how to develop loss models and decision models under uncertainty.\n\nTo illustrate this framework and the associated decision problems, let us consider an organisation which provides a service that is exposed to the threat of distributed denial-of-service (DDoS) attacks.\nSuch an organisation may opt to use network traffic filtering as well as a content distribution network, and may use multiple servers to balance the network load. \nAs such, there are various countermeasures for mitigating the threats of DDoS attacks and ensuring the availability of the service, but each of such self-mitigation measures adds to the upfront costs of risk reduction. \nTherefore, the organisation needs to determine the quantum of its security (i.e., risk reduction) budget and distribute this across these self-mitigation measures. We argue that this budget for risk reduction needs to and can be determined in tandem with risk transfer decisions, i.e., the purchase of cyber risk insurance.\nMoreover, a cyber risk insurer can incorporate incentive mechanisms in their insurance policies to encourage such risk reduction provision through offsetting the expenses of risk reduction by a discount in the pricing of the insurance premium.\nThis highlights the need for a comprehensive framework in which losses incurred by cyber threats can be realistically modelled and the rational decisions of organisations in the face of cyber threats can be quantitatively analysed.\n\n\n\nOur main contributions are as follows: \n\\begin{enumerate}\n    \\item We introduce the Bonus-Malus system to cyber risk insurance as a mechanism to provide incentive for the insured to adopt self-mitigation measures against cyber risk. \n\\item We develop a mathematical model of cyber losses and cyber risk insurance, and subsequently analyse the optimal cybersecurity provisioning process of the insured under the stochastic optimal control framework. \n\\item We develop an efficient algorithm based on dynamic programming to accurately solve the stochastic optimal control problem, under the assumption that the loss severity follows a truncated version of the g-and-h distribution. We also formally prove the correctness of the proposed algorithm. \n\\item We demonstrate through a numerical experiment that a properly designed cyber risk insurance contract with a Bonus-Malus system can resolve the issue of moral hazard, and can provide benefits for the insurer.\n\\end{enumerate}\n\nThe rest of the paper is organised as follows. \nSection~\\ref{sec:relatedwork} discusses related studies in the literature.\nIn Section~\\ref{sec:model}, we introduce the mathematical model of cyber losses and cyber risk insurance with a Bonus-Malus system. In Section~\\ref{sec:stocoptctr}, we present the optimal cybersecurity provisioning process and the dynamic programming algorithm. In Section~\\ref{sec:g-and-h}, we introduce the g-and-h distribution and use it as the model for loss severity. We present results from the numerical experiment in Section~\\ref{sec:exp}. Finally, Section~\\ref{sec:conclusion} concludes the paper. \n\n\\section{Related Work}\n\\label{sec:relatedwork}\nRecently, many studies analysed cyber risk insurance from a technology perspective. \nCybersecurity frameworks involving cyber risk insurance have been developed for specific IT systems, \nincluding computer networks \\citep{fahrenwaldt2018pricing,xu2019cybersecurity}, \nheterogeneous wireless network \\citep{lu2018cyber}, wireless cellular network \\citep{lu2018managing}, plug-in electric vehicles \\citep{hoang2017charging}, cloud computing \\citep{chase2017scalable}, and fog computing \\citep{feng2018evolving}. \n\nSome studies considered the interplay between self-mitigation measures (i.e., risk reduction) and cyber risk insurance, e.g., \\cite{pal2010analyzing,pal2014will,pal2017security,khalili2018designing,dou2020insurance}. \nThese studies investigated two important challenges in cyber risk insurance: risk interdependence and moral hazard. They found that in order to incentivise the insured to invest in self-mitigation measures, some form of contract discrimination, i.e., adjusting the insurance premium based on the insured's security investment, is necessary. \n\\cite{yang2014security,schwartz2014cyber,zhang2017bi} investigated these challenges in a networked environment, where cyber attacks can spread between neighbouring nodes, further complicating these challenges.\n\nThere are also studies which took the insured's perspective and analysed the security provisioning process using dynamic models. \n\\cite{chase2017scalable} developed a framework based on stochastic optimisation to jointly provision cyber risk insurance and cloud-based security services across multiple time periods in cloud computing applications. \n\\cite{zhang2018optimal} modelled the decisions on self-protections of the insured by a Markov decision process and investigated the problem of insurance contract design.\n\nA critical drawback of many of the existing studies is that they neglected the highly uncertain nature of losses incurred by cyber incidents. These studies relied on over-simplified assumptions, e.g., by modelling cyber loss as: a fixed amount \\citep{pal2010analyzing,pal2014will,yang2014security,hoang2017charging,feng2018evolving,dou2020insurance}, random with finite support \\citep{chase2017scalable,zhang2018optimal,lu2018managing}, or random with a simple parametric distribution \\citep{zhang2017bi,khalili2018designing}. These assumptions limit the practicality of these studies, and their results remain conceptual and non-applicable to realistic insurance loss modelling under a classical Loss Distribution Approach (LDA) framework \\citep{moscadelli2004modelling, peters2006bayesian, maillart2010heavy, shevchenko2013loss, eling2019actual, zeller2022comprehensive}.\nLDA is one of the most common modelling methods in the Advanced Measurement Approach (AMA) under the Basel II Accords. \nUnder the LDA framework, the probability distributions of loss severity (i.e., the impact of a single loss event) and annual loss frequency are modelled and estimated separately. The aggregate annual loss is thus modelled by a compound distribution. \nFor detailed discussions about LDA and a comparison with the Internal Measurement Approach, see \\citep{frachot2001loss} and the references therein.\nIn addition, see \\citep{dutta2006tale} for an evaluation of specific distributional assumptions in LDA and their impacts on the estimation of Operational Risk capital. \nIn our study about the design of cyber risk insurance contracts, we have built LDA into our cyber loss model and we have also incorporated a quantitative model for the effects of self-mitigation measures on cyber losses (see Section~\\ref{ssec:lossmodel} and Section~\\ref{ssec:mitigation}).\n\nMoreover, many of the aforementioned studies do not take into account the interplay between the upfront costs of risk prevention, the consequent reduced risks, and the possibility to exploit this interplay to design practical cyber risk insurance products.\nA review of cyber insurance product prospectus by major insurers, such as AIG's CyberEgde\\footnote{\\url{https:\/\/www.aig.com\/business\/insurance\/cyber-insurance\/}, accessed on 2020-12-10}, Allianz's Cyber Protect\\footnote{\\url{https:\/\/www.agcs.allianz.com\/solutions\/financial-lines-insurance\/cyber-insurance.html}, accessed on 2020-12-10}, and Chubb's Cyber Enterprise Risk Management (Cyber ERM)\\footnote{\\url{https:\/\/www.chubb.com\/us-en\/business-insurance\/cyber-enterprise-risk-management-cyber-erm.html}, \\linebreak accessed on 2020-12-10} also indicates that the insurance products in the market have yet to explicitly factor in the benefits of upfront risk reduction, or to offset those costs against that of risk transfer.\nFor the first time, we introduce the Bonus-Malus system which is frequently used in vehicle insurance products to address this gap in cyber risk insurance product design. \nIn vehicle insurance, Bonus-Malus systems are experience rating systems in which an insured who had one or more accidents is penalised by premium surcharges or \\textit{maluses} and an insured who had a claim-free year is rewarded with premium discounts or \\textit{bonuses} \\citep{lemaire1995bonus, neuhaus1988bonus}; see, e.g., \\citep{baione2002development, ragulina2017bonus, gomez2018multivariate, tzougas2018bonus} for discussions about the design and analysis of Bonus-Malus systems.\nA key characteristic of Bonus-Malus systems is the bonus hunger mechanism \\citep{holtan2001optimal, charpentier2017optimal, tzougas2018bonus}, i.e., under a Bonus-Malus system, an insured is willing to carry small losses themself in order to avoid premium surcharges in the future.\nOur optimal cybersecurity provisioning model in Section~\\ref{sec:stocoptctr} captures this aspect of Bonus-Malus systems by allowing the insured to decide whether to make a claim at the end of each policy year.\n\n\\section{Cyber Risk Insurance Policy and Bonus-Malus System}\n\\label{sec:model}\n\n\nLet us first present an overview of our cyber risk insurance model.\nTo begin, let us specify the frequency and severity model under the Loss Distribution Approach (LDA) that defines the financial loss process resulted from cyber loss events.\nWe consider $T\\in\\N$ consecutive years, and we assume that throughout each year $t$, the insured may suffer a random number ($N_t \\in \\N$) of cyber loss events arising from cyber attacks. Their loss amounts are denoted by $X^{(t)}_1,\\ldots,X^{(t)}_{N_t}$. The aggregate annual cyber loss in year $t$ is therefore $\\sum_{k=1}^{N_t}X^{(t)}_{k}$. \nThe insured has several choices to attempt to mitigate these cyber loss events and reduce the risk, including enhancing the security and resilience of their IT infrastructure and reserving Tier~I capital to cover the incurred losses.\nIn regards to the internal IT infrastructure, it will be assumed that the insured has the option to adopt a self-mitigation measure that can reduce the severity of cyber loss events up to a fixed amount of loss. \nIn addition, the insured can choose to purchase a cyber risk insurance policy which gives the insured the right to claim the aggregate cyber loss incurred, up to a maximum cap imposed by the insurance contract, in an agreed interval of time (typically annually), minus a deductible. \nPlease refer to Table~\\ref{tab:model-notations} for notations used in the cyber risk insurance model.\n\n\\begin{table}\n\\begin{center}\n\\caption{Notations in the cyber risk insurance model}\n\\label{tab:model-notations}\n\\vspace{0.5em}\n\\begin{tabular}{ll}\n$T$ & number of policy years \\\\\n$N_t$ & number of cyber loss events in year $t$ \\\\\n$X^{(t)}_1,\\ldots,X^{(t)}_{N_t}$ & loss amounts of the cyber loss events in year $t$  \\\\\n$\\mathcal{W}$ & space representing annual loss frequency and severities \\\\ \n$(\\Omega,\\mathcal{F}_T,\\PROB,(\\mathcal{F}_t)_{t=0:T})$ & filtered probability space \\\\ \n$W_t$ & loss frequency and severities in year $t$ \\\\\n$\\psi_N(\\cdot)$ & probability generating function of the loss frequency \\\\ \n$F_X(\\cdot)$ & distribution function of the loss severity \\\\ \n$\\mathcal{D}$ & set representing self-mitigation measures \\\\\n$\\beta(d)$ & annual investment required by the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$\\gamma(d)$ & loss reduction effect of the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$L(d,w)$ & aggregate annual cyber loss with the self-mitigation measure $d\\in\\mathcal{D}$ \\\\\n$\\delta_{\\mathrm{i}\\mathrm{n}}(t)$ & initial sign-on fee of cyber risk insurance in year $t$ \\\\\n$\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)$ & withdrawal penalty of cyber risk insurance in year $t$ \\\\\n$\\delta_{\\mathrm{r}\\mathrm{e}}$ & re-activation penalty of cyber risk insurance \\\\\n$\\mathcal{B}$ & set representing Bonus-Malus levels \\\\\n$\\mathcal{I}$ & set representing states of the cyber risk insurance contract \\\\\n$\\mathcal{B}\\mathcal{M}(\\cdot,\\,\\cdot)$ & transition rules of the Bonus-Malus system \\\\\n$\\mathcal{B}\\mathcal{M}_0(\\cdot,\\,\\cdot)$ & transition rules of the Bonus-Malus system for inactive contract \\\\\n$p^{\\mathcal{B}\\mathcal{M}}(b,t)$ & insurance premium in Bonus-Malus level $b$ in year $t$ \\\\\n$l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,t)$ & insurance deductible in Bonus-Malus level $b$ in year $t$ \\\\\n$l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)$ & maximum insurance compensation in Bonus-Malus level $b$ in year $t$ \\\\\n$\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,l)$ & insurance compensation in Bonus-Malus level $b$ in year $t$ with loss $l$\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Cyber Loss Model}\n\\label{ssec:lossmodel}\n\nLet us define $\\mathcal{W}:=\\bigcup_{n\\in\\Z_+}\\{n\\}\\times\\R^n_+$ to be the space representing all possible combinations of realisations of the number of cyber loss events per year (frequency) and individual loss amounts per event (severities).\nLet $\\mathfrak{B}(\\mathcal{W}):=\\sigma\\big(\\bigcup_{n\\in\\Z_+}\\{(n,B):B\\in\\mathfrak{B}(\\R^n_+)\\}\\big)$ be a $\\sigma$-algebra on $\\mathcal{W}$, where $\\mathfrak{B}(\\R^n_+)$ denotes the Borel subsets of $\\R^n_+$. \nLet us consider the space $\\Omega:=(\\mathcal{W})^T=\\underbrace{\\mathcal{W}\\times\\cdots\\times\\mathcal{W}}_{T \\text{ times}}$. For each $\\omega=(w_1,\\ldots,w_T)=\\big(\\big(n_1,x^{(1)}_1,\\ldots,x^{(1)}_{n_1}\\big),\\ldots,$ $\\big(n_T,x^{(T)}_1,\\ldots,x^{(T)}_{n_T}\\big)\\big)\\in\\Omega$, we define $W_t(\\omega):=w_t$ and $N_t(\\omega):=n_t$ for $t=1,\\ldots,T$. \nLet $\\PROB_1$ be a probability measure on $(\\mathcal{W},\\mathfrak{B}(\\mathcal{W}))$, where the subscript ``1'' indicates that it is a probability measure for the cyber loss events occurring in a single year.\nLet $\\PROB:=\\underbrace{\\PROB_1\\otimes\\cdots\\otimes\\PROB_1}_{T\\text{ times}}$, and let $(\\mathcal{F}_t)_{t=0:T}$ be a filtration on $\\Omega$, defined by \n$\\mathcal{F}_0:=\\{\\emptyset,\\Omega\\}$, $\\mathcal{F}_t:=\\sigma((W_s)_{s=1:t})$. Then, $(\\Omega,\\mathcal{F}_{T},\\PROB,(\\mathcal{F}_t)_{t=0:T})$ is a filtered probability space. Under these definitions, $W_1,\\ldots,W_T$ are independently and identically distributed ({i.i.d.})\\ random variables. \nLet $\\psi_{N}(s):=\\EXP[s^{N_1}]$ denote the probability generating function (pgf) of the loss frequency distribution. We assume that for $t=1,\\ldots,T$, \n\\begin{align}\n\\begin{split}\n&\\phantom{=}\\;\\;\\PROB\\Big[\\Big\\{\\omega=\\big(\\big(n_1,x^{(1)}_1,\\ldots,x^{(1)}_{n_1}\\big),\\ldots,\\big(n_T,x^{(T)}_1,\\ldots,x^{(T)}_{n_T}\\big)\\big):n_t=n,\\;x^{(t)}_k\\le z_k,\\forall 1\\le k\\le n\\Big\\}\\Big]\\\\\n&=\\PROB[N_t=n]\\prod_{k=1}^nF_X(z_k),\n\\end{split}\n\\end{align}\nwhere $F_X(\\cdot)$ is the distribution function of the severity distribution. This implies that given the loss frequency in a year, the individual loss amounts in that year are i.i.d. We assume that the severity distribution has finite expectation, i.e., $\\int_{\\R_+}|x|\\DIFFM{F_X}{\\DIFF x}<\\infty$. \nFor convenience, we write $W_t=\\big(N_t,X^{(t)}_1,\\ldots,X^{(t)}_{N_t}\\big)$. \nWe use $W$ to denote a random variable that has the same distribution as $W_1,\\ldots,W_T$. \nSimilarly, we use $N$ to denote a random variable that has the same distribution as $N_1,\\ldots,N_T$, and we use $X$ to denote a random variable that has the distribution function $F_X$. \n\n\\subsection{Self-Mitigation Measures}\n\\label{ssec:mitigation}\n\nLet us assume that there exists $D\\in\\N$ different self-mitigation measures, and the insured makes the decision to either adopt one of the self-mitigation measures or to not adopt any self-mitigation measure at the beginning of each year. The self-mitigation measure $d\\in\\mathcal{D}:=\\{0,1,\\ldots,D\\}$ requires an annual investment of $\\beta(d)\\in\\R_+$ per year, and reduces the severity of each cyber loss by up to $\\gamma(d)\\in\\R_+$, that is, the severity of a loss will be decreased from $X$ to\\footnote{Throughout the paper, we use the following notations: $(x)^+:=\\max\\{x,0\\}$, $x\\vee y:=\\max\\{x,y\\}$ and $x\\wedge y:=\\min\\{x,y\\}$.} $\\left(X-\\gamma(d)\\right)^+$ with the adoption of the self-mitigation measure $d$. We assume that $\\beta(0)=\\gamma(0)=0$. \nThus, if the insured decides to adopt the self-mitigation measure $d$ in a year, then the total loss suffered by the insured that year is given by\n\\begin{align}\nL(d,w):=\\sum_{k=1}^{n}\\left(x_k-\\gamma(d)\\right)^+,\n\\label{eqn:yearlyloss}\n\\end{align}\nwhen the corresponding loss frequency and severities that year are $w=(n,x_1,\\ldots,x_n)\\in\\mathcal{W}$.\n\n\n\\subsection{Cyber Risk Insurance Policy}\n\\label{ssec:cyberinsurance}\nLet us now consider a cyber risk insurance contract that lasts for $T$ years. \nAt the beginning of each year, the insured decides whether to activate the contract. In the case that the contract has been activated in a previous year, this corresponds to the insured deciding whether to continue the contract. If the contract is activated, the insured pays a premium to the insurer at the start of the year, in exchange for insurance coverage throughout the year. If the insured decides to withdraw from the contract, they no longer pays the premium and the contract is deactivated so that the insured receives no coverage. \nWe further assume that the insured pays the insurer an initial sign-on fee for fixed costs and contract origination the first time a contract is initiated, in addition to the premium, and that this amount varies deterministically over time and will be denoted by $\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\ge 0$ in year $t$. This can be used to incentivise the insured to activate the contract early. \nFurthermore, we also assume that the insured pays the insurer a deterministic and time-dependent penalty, denoted by $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\ge0$, when they withdraws from the contract in year $t$. Once withdrawn, the insured may re-activate the contract in a later year with a fixed penalty $\\delta_{\\mathrm{r}\\mathrm{e}}\\ge0$. \n\nSuppose that the aggregate cyber loss suffered by the insured in a year is $L$. At the end of the year, the insured decides whether to make a claim to the insurer. Once the claim is processed, the insured receives a payment of $(L-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}})^+\\wedge l_{\\max}$ from the insurer as compensation, that is, the insured covers the loss up to the deductible $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}\\ge 0$, and the insurer covers all of the remaining loss up to the maximum compensation $l_{\\max}\\ge0$. \n\n\\subsection{Bonus-Malus System}\n\\label{ssec:bonusmalus}\n\nWe now introduce the Bonus-Malus system to cyber risk insurance contracts. Let us assume that there are $B\\in\\N$ Bonus-Malus levels in the contract, denoted by $\\mathcal{B}:=\\{-\\underline{B},\\ldots,-1,0,1,\\ldots,\\overline{B}\\}$, where $\\underline{B}+\\overline{B}+1=B$. \nHere, the level $0$ is the initial Bonus-Malus level, and the lower the Bonus-Malus level, the higher the experience rating.\nConsequently, a negative Bonus-Malus level grants the insured premium discounts, while a positive Bonus-Malus level penalises the insured by premium surcharges.\nAt $t=0$, the insured starts in the initial Bonus-Malus level, denoted by $b_0=0$. At the end of the $t$-th year, given that the contract is still active, the insurer determines the Bonus-Malus level of the insured based on their previous level $b_{t-1}$ and the amount of insurance claim $C_t$ that was given out to the insured in the $t$-th year, that is, $b_{t}=\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$, where $\\mathcal{B}\\mathcal{M}:\\mathcal{B}\\times\\R_+\\to\\mathcal{B}$ denotes the deterministic rules that are transparent to the insured at the signing of the contract. We make the assumption that $\\mathcal{B}\\mathcal{M}(b,C)$ is non-decreasing in $C$ for each $b\\in\\mathcal{B}$. \nEven when the insured has withdrawn from the contract, we assume that their Bonus-Malus level is still updated annually. \nConcretely, let us define $\\mathcal{I}:=\\{\\mathrm{n}\\mathrm{o},\\mathrm{o}\\mathrm{n},\\mathrm{o}\\mathrm{f}\\Tf_1,\\ldots,\\mathrm{o}\\mathrm{f}\\Tf_{T}\\}$ as the set of all possible states of the cyber risk insurance contract. In $\\mathcal{I}$, ``$\\mathrm{n}\\mathrm{o}$'' denotes that the contract has not been signed yet, ``$\\mathrm{o}\\mathrm{n}$'' denotes that the contract is active, and ``$\\mathrm{o}\\mathrm{f}\\Tf_y$'' denotes that the contract is withdrawn where $y\\in\\Z_+$ is a counter variable that is updated annually as long as the insured does not re-activate the contract. \nLet $\\mathcal{B}\\mathcal{M}_{0}:\\mathcal{B}\\times\\mathcal{I}\\to\\mathcal{B}\\times\\mathcal{I}$ be a deterministic transition function that represents the update rules after the insured withdraws from the contract. At the end of the $t$-th year, given that the contract is inactive, the insurer determines the Bonus-Malus level $b_t$ and the insurance state $i_t$ of the insured based on their Bonus-Malus level and the insurance state in the previous year, that is, $(b_t,i_t)=\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$. Since no such update is possible before the insured activates the contract, it is required that $\\mathcal{B}\\mathcal{M}_{0}(b,\\mathrm{n}\\mathrm{o})=(b,\\mathrm{n}\\mathrm{o})$ for all $b\\in\\mathcal{B}$. We will formally model the evolution of $(b_t,i_t)_{t=0:T}$ by a controlled stochastic process in Section~\\ref{ssec:provisionprocess}.\n\nWith the addition of the Bonus-Malus system to the cyber risk insurance contract, we assume that the premium depends on both time and the current Bonus-Malus level of the insured, and is given by $p^{\\mathcal{B}\\mathcal{M}}(b,t)$, where $p^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$ is a deterministic function that is increasing in the first argument. The deductible and the maximum compensation are also assumed to be dependent on both time and the Bonus-Malus level, and are given by deterministic functions $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$ and $l_{\\max}^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\to\\R_+$. We define the function $\\lambda^{\\mathcal{B}\\mathcal{M}}:\\mathcal{B}\\times\\{1,\\ldots,T\\}\\times\\R_+\\to\\R_+$ by\n\\begin{align}\n\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,l):=(l-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}^{\\mathcal{B}\\mathcal{M}}(b,t))^+\\wedge l_{\\max}^{\\mathcal{B}\\mathcal{M}}(b,t)\n\\end{align}\nto simplify the notation for the claimable loss from the insurer.  \n\n\\begin{remark}\nOur loss model and insurance model are particularly suitable for analysing cyber risk insurance, due to the nature of cyber loss events.\nFirstly, the compound loss model in LDA is vital for capturing the low frequency and high impact nature of certain cyber loss events \\citep{maillart2010heavy, biener2015insurability, eling2017data, eling2015modelling, eling2019actual}. \nIn particular, this loss model allows us to use the highly flexible g-and-h distribution for the loss severity, in order to capture the right-skewness and the heavy-tail of cyber losses.\nThis provides adequate modelling flexibility so that one can tailor this loss model, particularly the skewness and the tail-index of the loss distribution, according to specific kinds of cyber threats and different business lines. \nWe will discuss the details of the g-and-h distribution in Section~\\ref{sec:g-and-h}.\nSecondly, as studies such as \\citep{oughton2019stochastic, armenia2021dynamic} have discussed, despite that complete mitigation of cyber risk is impractical, prevention of crippling damages from cyber attacks, which often include secondary damages to downstream customers, can be achieved with relatively low cybersecurity expenses. \nFor example, countermeasures such as the adoption of two-factor authentication and the regular update or reconfiguration of the software system \\citep{pate2018cyber} can often result in significant reduction of the risk of intrusion. \nMoreover, the adoption of self-mitigation measures generates positive externalities since it improves the overall security of the cyberspace. \nDue to these factors, there is a strong motive for cyber risk insurers to offer incentive mechanisms in their insurance products to encourage such upfront risk reduction effort and to alleviate the problem of moral hazard.\nOverall, the combination of this flexible loss model and the incentive mechanisms built into the Bonus-Malus system is novel in the context of cyber risk insurance. \nOn the other hand, the i.i.d.\\ assumption on the individual loss amounts in our model means that it does not account for the systemic aspect of cyber risk.\nWe would also like to remark that the general specifications of our cyber risk insurance model make it also applicable to other idiosyncratic risk types which can be partially prevented by self-mitigation measures. \nDespite that, our analyses and discussions about our model will be carried out in the context of cyber risk, and we will not generalise our model to other risk types in order not to obscure the main objective of this paper.\n\\label{rmk:cyber-specificity}\n\\end{remark}\n\n\n\\section{Optimal Cybersecurity Provisioning and Stochastic Optimal Control}\n\\label{sec:stocoptctr}\n\n\\subsection{Cybersecurity Provisioning Process}\n\\label{ssec:provisionprocess}\nNow, having introduced the model for cyber losses and cyber risk insurance, we consider the problem of optimal cybersecurity provisioning from the insured's point of view. \nIt is assumed that the cybersecurity provisioning process takes place for $T$ consecutive years (same as the length of the cyber risk insurance contract).\nBefore the first year, the state of the cyber risk insurance contract is initialised to ``$\\mathrm{n}\\mathrm{o}$'', i.e., $i_0=\\mathrm{n}\\mathrm{o}$, and the Bonus-Malus level is initialised to level~$0$, i.e., $b_0=0$.\nSubsequently, each year $t\\in\\{1,\\ldots,T\\}$ consists of the three following stages:\n\\begin{enumerate}\n\\item \\textbf{Provision Stage.} The insured decides: (i) the self-mitigation measure to adopt in this year, denoted by $d_t\\in\\mathcal{D}$ and (ii) whether to activate\/withdraw\/re-activate the cyber risk insurance contract, denoted by $\\iota_t\\in\\{0,1\\}$. Then, depending on the decision $\\iota_t$, a premium payment $p^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t)$ and\/or a sign-on fee $\\delta_{\\mathrm{i}\\mathrm{n}}(t)$, a withdraw penalty $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)$, or a re-activation penalty $\\delta_{\\mathrm{r}\\mathrm{e}}$ will be incurred, as stated in Section~\\ref{ssec:cyberinsurance} and Section~\\ref{ssec:bonusmalus}.\n\n\\item \\textbf{Operation Stage.} The random cyber loss events and the corresponding cyber losses suffered by the insured in this year, denoted by $W_t$, is realised in this stage according to the model described in Section~\\ref{ssec:lossmodel} and Section~\\ref{ssec:mitigation}. The aggregate cyber loss incurred is given by $L(d_t,W_t)$.\n\n\\item \\textbf{Claim Stage.} If the insurance contract is active, i.e., $\\iota_t=1$, the insured decides whether or not to make a claim, denoted by $j_t\\in\\{0,1\\}$. In the case where a claim is made, i.e., $j_t=1$, the insured receives a compensation from the insurer according to the aggregate cyber loss that year, given by $\\lambda^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t,L(d_t,W_t))$. Subsequently, the cyber risk insurance contract state $i_t$ and the Bonus-Malus level $b_t$ are updated based on the decisions $\\iota_t$ and $j_t$, as described in Section~\\ref{ssec:bonusmalus}. \nConcretely, if $\\iota_t=1$, then the insurance contract state will be ``$\\mathrm{o}\\mathrm{n}$'' and the Bonus-Malus level will be updated according to the function $\\mathcal{B}\\mathcal{M}(\\cdot,\\,\\cdot)$, i.e., $i_t=\\mathrm{o}\\mathrm{n}$,  $b_t=\\mathcal{B}\\mathcal{M}\\big(b_{t-1},j_t\\lambda^{\\mathcal{B}\\mathcal{M}}(b_{t-1},t,L(d_t,W_t))\\big)$.\nIf $\\iota_t=0$, then the insurance contract state and the Bonus-Malus level will be updated according to the function $\\mathcal{B}\\mathcal{M}_0(\\cdot,\\,\\cdot)$, i.e., $(b_t,i_t)=\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$.\n\\end{enumerate}\n\n\\begin{table}\n\\begin{center}\n\\caption{Notations in the optimal cybersecurity provisioning process and stochastic optimal control}\n\\label{tab:dp-notations}\n\\vspace{0.5em}\n\\begin{tabular}{ll}\n$d_t$ & decision of self-mitigation measure adopted in year $t$ \\\\\n$\\iota_t$ & decision to activate\/withdraw\/re-activate the insurance contract in year $t$ \\\\\n$j_t$ & decision of whether to make a claim in year $t$  \\\\\n$\\Pi$ & set containing all admissible decision policies \\\\ \n$f_t(b,i,d,\\iota,j,w)$ & state transition function in year $t$ in stochastic optimal control \\\\ \n$g_t(b,i,d,\\iota,j,w)$ & cost function in year $t$ in stochastic optimal control \\\\\n$(b^\\pi_t,i^\\pi_i)_{t=0:T}$ & controlled stochastic process representing the Bonus-Malus and insurance states \\\\ \n$e^{-r}$ & discount factor in stochastic optimal control \\\\\n$V^\\pi_t$ & expected discounted future cost at year $t$ in stochastic optimal control \\\\ \n$\\mathbb{V}_0$ & the optimal value of the stochastic optimal control problem \\\\\n$\\mathcal{V}_t(b,i)$ & value function in dynamic programming \\\\\n$\\widehat{d}_t(b,i)$ & one-stage optimal decision of self-mitigation measure in year $t$ \\\\\n$\\widehat{\\iota}_t(b,i)$ & one-stage optimal decision of cyber risk insurance in year $t$ \\\\\n$\\widehat{j}_t(b,i,w)$ & one-stage optimal decision of insurance claim in year $t$ \\\\\n$P^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]$ & transition kernel of the optimally controlled process \\\\\n$\\overline{P}^\\star_t(b,i)$ & marginal state occupancy probability of the optimally controlled process \\\\\n$\\zeta^{(m)}_t(b,i,w)$ & a quantity of interest that depends on the state and the losses \\\\\n$\\overline{\\zeta}^{(m)}_t$ & expected value of a quantity of interest \\\\\n$\\overline{Z}_{\\zeta^{(m)}}$ & aggregate expected value of a quantity of interest\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\nNext, in order to formally define an optimal decision policy, let us introduce the stochastic optimal control formulation of the cybersecurity provisioning process. Please refer to Table~\\ref{tab:dp-notations} for notations used in the cybersecurity provisioning process and the stochastic optimal control formulation.\nLet\n\\begin{align}\n\\begin{split}\n\\Pi:=\\Big\\{\\pi=(d_t,\\iota_t,&j_t)_{t=1:T}:d_t:\\Omega\\to\\mathcal{D},\\;\\iota_t:\\Omega\\to\\{0,1\\} \\text{ are }\\mathcal{F}_{t-1}\\text{-measurable},\\\\\n&j_t:\\Omega\\to\\{0,1\\} \\text{ is }\\mathcal{F}_t\\text{-measurable},\\;\\{\\iota_t=0,j_t=1\\}=\\emptyset,\\text{ for }t=1,\\ldots,T\\Big\\}\n\\end{split}\n\\end{align}\ndenote the set of all admissible decision policies. The conditions in the above definition are explained as follows.\n\\begin{itemize}\n\\item The decisions $d_t$ and $\\iota_t$ are made before observing $W_t$, hence may depend on all available information up to year $t-1$.\n\\item The decision $j_t$ is made after observing $W_t$, hence may depend on all available information up to year $t$.\n\\item The condition $\\{\\iota_t=0,j_t=1\\}=\\emptyset$ requires that the insured may only make a claim (i.e., $j_t=1$) when the insurance contract is activated (i.e., $\\iota_t=1$) in year $t$. \n\\end{itemize}\nFor $t=1,\\ldots,T$, let $f_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{D}\\times\\{0,1\\}\\times\\{0,1\\}\\times\\mathcal{W}\\to\\mathcal{B}\\times\\mathcal{I}$ be the state transition function for each year $t$, given by\n\\begin{align}\n\\begin{split}\nf_t(b,i,d,\\iota,j,w):=\\begin{cases}\n\\big(\\mathcal{B}\\mathcal{M}(b,j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))),\\mathrm{o}\\mathrm{n}\\big) &\\text{if }\\iota=1,\\\\\n\\mathcal{B}\\mathcal{M}_0(b,i) &\\text{if }\\iota=0,\n\\end{cases}\n\\end{split}\n\\label{eqn:socstate}\n\\end{align}\nthat is, $f_t(b,i,d,\\iota,j,w)$ returns the Bonus-Malus level and the insurance state in year $t$ given that the Bonus-Malus level and the insurance state in year $t-1$ are $b$ and $i$, the decisions in year $t$ are $d$, $\\iota$, $j$, and the cyber loss events in year $t$ are $w$.\nFor $t=1,\\ldots,T$, let $g_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{D}\\times\\{0,1\\}\\times\\{0,1\\}\\times\\mathcal{W}\\to\\R_+$ be the cost function for each year $t$, given by \n\\begin{align}\n\\begin{split}\ng_t(b,i,d,\\iota,j,w)&:=\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}\\\\\n&\\hspace{1.5cm}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+L(d,w)-\\iota j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w)),\n\\end{split}\n\\label{eqn:soccost}\n\\end{align}\nwhere $\\beta(d)$ is the investment of adopting the self-mitigation measure $d$, $\\iota p^{\\mathcal{B}\\mathcal{M}}(b)$ corresponds to the cyber risk insurance premium, $\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}$ corresponds to the sign-on\/withdrawal\/re-activation costs, $L(d,w)$ is the aggregate cyber loss, and $\\iota j\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))$ corresponds to the compensation from the insurer. \n\nSubsequently, for any decision policy $\\pi=(d_t,\\iota_t,j_t)_{t=1:T}\\in\\Pi$, let us define the $(\\mathcal{F}_t)_{t=0:T}$-adapted controlled stochastic process $\\big(b^{\\pi}_t,i^{\\pi}_t\\big)_{t=0:T}$ as follows:\n\\begin{align}\n\\begin{split}\n(b^\\pi_0,i^\\pi_0)&:=(0,\\mathrm{n}\\mathrm{o}),\\\\\n(b^\\pi_t,i^\\pi_t)&:=f_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)\\quad\\text{for }t=1,\\ldots,T.\n\\end{split}\n\\label{eqn:markovprocess}\n\\end{align}\nThen, $g_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)$ corresponds to the cybersecurity cost in year $t$. For $\\pi\\in\\Pi$, and $t=1,\\ldots,T$, define $V^{\\pi}_t$ by\n\\begin{align}\nV^\\pi_t:=\\EXP\\Big[\\textstyle\\sum_{s=t+1}^Te^{-(s-t)r}g_s(b^\\pi_{s-1},i^\\pi_{s-1},d_s,\\iota_s,j_s,W_s)\\Big|\\mathcal{F}_{t}\\Big],\n\\end{align}\nwhere $0<e^{-r}\\le 1$ is the discount factor and $V^\\pi_T:=0$,\nthat is, $V^\\pi_t$ is the expected value of the discounted total future cybersecurity cost viewed at the end of year $t$.\n\nWe assume that in the optimal cybersecurity provisioning process, the objective of the insured is to minimise the expected value of the discounted total cybersecurity cost, i.e., $V^\\pi_0$, over all admissible decision policies $\\pi\\in\\Pi$. This is formulated as the following finite horizon stochastic optimal control problem:\n\\begin{align}\n\\mathbb{V}_0:=\\inf_{\\pi\\in\\Pi}V^\\pi_0=\\inf_{\\pi\\in\\Pi}\\EXP\\Big[\\textstyle\\sum_{t=1}^Te^{-tr}g_t(b^\\pi_{t-1},i^\\pi_{t-1},d_t,\\iota_t,j_t,W_t)\\Big]. \n\\label{eqn:stocoptctr}\n\\end{align}\n\n\n\\subsection{Dynamic Programming Algorithm}\n\\label{ssec:dpalgo}\nThe stochastic optimal control problem in (\\ref{eqn:stocoptctr}) can be solved efficiently via the dynamic programming algorithm. \nThe dynamic programming algorithm iteratively solves one-stage optimisation problems while going backward in time.\nIn the following, let us denote the optimal values of these one-stage optimisation problems by $(\\mathcal{V}_t)_{t=0:T}$, and denote their corresponding optimisers by $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)_{t=1:T}$.\nFor $t=T,T-1,\\ldots,0$, let $\\mathcal{V}_{t}:\\mathcal{B}\\times\\mathcal{I}\\to\\R_+$ be recursively defined as follows: for every $b\\in\\mathcal{B},i\\in\\mathcal{I}$, let\n\\begin{align}\n\\begin{split}\n\\mathcal{V}_T(b,i)&:=0,\\\\\n\\mathcal{V}_{t-1}(b,i)&:=e^{-r}\\min_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}}\\bigg\\{\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\bigg\\}\\\\\n&\\hspace{280pt}\\text{for }t=T,T-1,\\ldots,1.\n\\end{split}\n\\label{eqn:dpvalue}\n\\end{align}\nMoreover, for $t=1,\\ldots,T$, let $\\widehat{d}_t:\\mathcal{B}\\times\\mathcal{I}\\to\\mathcal{D}$, $\\widehat{\\iota}_t:\\mathcal{B}\\times\\mathcal{I}\\to\\{0,1\\}$, and $\\widehat{j}_t:\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}\\to\\{0,1\\}$ be defined as follows: for every $b\\in\\mathcal{B},i\\in\\mathcal{I}$, let\n\\begin{align}\n\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)&\\in \\argmin_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}}\\bigg\\{\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\bigg\\},\\label{eqn:dpconcept-d-iota}\\\\\n\\widehat{j}_t(b,i,w)&:=\\begin{cases}\n1 & \\text{if }\\widehat{\\iota}_t(b,i)=1,\\;\\Big[g_{t}(b,i,\\widehat{d}_t(b,i),1,1,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,\\widehat{d}_t(b,i),1,1,w)\\big)\\Big] \\\\\n&\\hspace{2.5cm}< \\Big[g_{t}(b,i,\\widehat{d}_t(b,i),1,0,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,\\widehat{d}_t(b,i),1,0,w)\\big)\\Big],\\\\\n 0 & \\text{otherwise}.\n\\end{cases}\n\\label{eqn:dpconcept-j}\n\\end{align}\nThus, for $t=0,\\ldots,T$, $\\mathcal{V}_t(b,i)$ corresponds to the optimal total discounted future cybersecurity cost when viewing the optimal cybersecurity provisioning process at the end of year $t$ when the Bonus-Malus level and the insurance state are $b$ and $i$. \nObserve from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) that $\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)$ solves the minimisation outside the expectation in (\\ref{eqn:dpvalue}), while $\\widehat{j}_t(b,i,W)$ solves the minimisation inside the expectation in (\\ref{eqn:dpvalue}) with $(d,i)\\leftarrow\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i)\\big)$.\nSince the minimisation over $j\\in\\{0,1\\}$ in (\\ref{eqn:dpvalue}) is inside the expectation, the decision $j$ is made after observing the realisation of the cyber loss events $W$, and thus the optimiser $\\widehat{j}_t$ is allowed to depend on $W$. On the other hand, since the minimisation over $d\\in\\mathcal{D}$, $\\iota\\in\\{0,1\\}$ is outside the expectation, these decisions need to be made before observing $W$, and thus $\\widehat{d}_t$ and $\\widehat{\\iota}_t$ cannot depend on $W$.\nIn the first iteration of the dynamic programming algorithm, one computes $\\big(\\widehat{d}_T(b,i),\\widehat{\\iota}_T(b,i),\\widehat{j}_T(b,i,w)\\big)$ from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) for every $(b,i,w)\\in\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}$. Moreover, one computes the values of $(\\mathcal{V}_{T-1}(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ from (\\ref{eqn:dpvalue}).\nIn subsequent iterations of the dynamic programming algorithm, one iterates through $t=T-1,T-2,\\ldots,1$, where, in each iteration, one considers the one-stage optimisation problem in year $t$ given the values $(\\mathcal{V}_t(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$.\nSpecifically, one computes $\\big(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i),\\widehat{j}_t(b,i,w)\\big)$ from (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}) for every $(b,i,w)\\in\\mathcal{B}\\times\\mathcal{I}\\times\\mathcal{W}$, and computes the values of $(\\mathcal{V}_{t-1}(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ from (\\ref{eqn:dpvalue}).\nThe following theorem shows how one can construct an optimal decision policy $\\pi^\\star$ for the stochastic optimal control problem (\\ref{eqn:stocoptctr}) by piecing together the one-stage optimisers $\\big(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t\\big)_{t=1:T}$ obtained via dynamic programming.\n\n\n\\begin{theorem}\nLet the functions $(\\mathcal{V}_t)_{t=0:T}$ be defined as in (\\ref{eqn:dpvalue}), let the functions $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)_{t=1:T}$ be defined as in (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}), and let $\\pi^\\star=(d^\\star_t,\\iota^\\star_t,j^\\star_t)_{t=1:T}\\in\\Pi$ be recursively defined as follows: \n\\begin{align}\n\\begin{split}\n&\\hspace{-0.75cm}\\big(b^{\\pi^\\star}_{0},i^{\\pi^\\star}_{0}\\big):=(0,\\mathrm{n}\\mathrm{o}),\\\\\n\\text{for }t&=1,\\ldots,T,\\text{ let:}\\\\\n&d^\\star_t:=\\widehat{d}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star}\\big),\\quad\\quad \\iota^\\star_t:=\\widehat{\\iota}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star}\\big),\\quad\\quad j^\\star_t:=\\widehat{j}_t\\big(b_{t-1}^{\\pi^\\star},i_{t-1}^{\\pi^\\star},W_t\\big).\n\\end{split}\n\\label{eqn:dpconcept-opt-process}\n\\end{align}\nThen, the following holds:\n\\begin{align}\n\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=V_0^{\\pi^\\star}=\\mathbb{V}_0.\n\\end{align}\n\\label{thm:dp}\n\\end{theorem}\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\nBesides an optimal decision policy $\\pi^\\star$ constructed in Theorem~\\ref{thm:dp}, we are often also interested in other quantities related to the optimal cybersecurity provisioning process, such as the transition kernels and the marginal state occupancy probabilities of the process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ as well as other quantities of interest.\nSpecifically, given an optimal decision policy $\\pi^\\star$ constructed in Theorem~\\ref{thm:dp}, its associated optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ is a discrete-time time-inhomogeneous Markov chain, and we are interested in its transition kernels $\\big(P^\\star_t\\big)_{t=1:T}$, where\n\\begin{align}\n\\begin{split}\nP^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]&:=\\PROB\\Big[\\big(b^{\\pi^\\star}_{t},i^{\\pi^\\star}_{t}\\big)=(b',i')\\Big|\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1}\\big)=(b,i)\\Big] \\\\\n&\\hspace{160pt}\\forall (b,i),(b',i')\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=1,\\ldots,T,\n\\end{split}\n\\label{eqn:dpalgo-kernel}\n\\end{align}\nas well as its marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, where\n\\begin{align}\n\\overline{P}^\\star_t(b,i):=\\PROB\\Big[\\big(b^{\\pi^\\star}_{t},i^{\\pi^\\star}_{t}\\big)=(b,i)\\Big]\\qquad\\forall (b,i)\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=0,\\ldots,T.\n\\label{eqn:dpalgo-marg}\n\\end{align}\nUsing these probabilities, we are able to compute important quantities that will aid us in the design and evaluation of cyber risk insurance contracts, such as the expected number of years the insured will spend in each of the Bonus-Malus levels.\nMoreover, we are often interested in computing the expected values of other quantities of interest which can be used as evaluation metrics for the cyber risk insurance contract.\nFormally, suppose that we are interested in $M$ quantities, and let $\\zeta^{(m)}_t:\\mathcal{B}\\times\\mathcal{I}\\times\\R_+\\to\\R$ for $t=1,\\ldots,T$, $m=1,\\ldots,M$ satisfy\n\\begin{align}\n\\EXP\\big[\\big|\\zeta^{(m)}_t(b,i,W)\\big|\\big]<\\infty\\qquad \\forall (b,i)\\in\\mathcal{B}\\times\\mathcal{I},\\;\\text{for }t=1,\\ldots,T,\\;\\text{for }m=1,\\ldots,M.\n\\label{eqn:qoi}\n\\end{align}\nLet us define\n\\begin{align}\n\\overline{\\zeta}^{(m)}_t&:=\\EXP\\big[\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)\\big]\\qquad\\text{for }t=1,\\ldots,T,\\;\\text{for }m=1,\\ldots,M,\\label{eqn:qoi-expval}\\\\\n\\overline{Z}_{\\zeta^{(m)}}&:=\\sum_{t=1}^T\\overline{\\zeta}_t^{(m)}\\hspace{139pt}\\qquad\\text{for }m=1,\\ldots,M.\\label{eqn:qoi-aggregate}\n\\end{align}\nThus, for $m=1,\\ldots,M$, $\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)$ represents a quantity of interest in year $t$ that (possibly) depends on the Bonus-Malus level and the insurance state at the end of year $t-1$, i.e., $b^{\\pi^\\star}_{t-1}$ and $i^{\\pi^\\star}_{t-1}$, and the losses in year $t$, i.e., $W_t$. \nMoreover, for $m=1,\\ldots,M$, $\\overline{\\zeta}^{(m)}_t$ denotes the expected value of the quantity of interest $\\zeta^{(m)}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1},W_t\\big)$ in year $t$, and $\\overline{Z}_{\\zeta^{(m)}}$ denotes the aggregate expected value of this quantity of interest over $T$ years. \nDue to the independence between $\\mathcal{F}_{t-1}$ and $\\sigma(W_t)$, one observes that $(\\overline{\\zeta}^{(m)}_t)_{t=1:T}$ and hence also $\\overline{Z}_{\\zeta^{(m)}}$ can be computed based on the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}). Example~\\ref{exp:quantities} below provides concrete examples of quantities of interest that can aid the design and analysis of cyber risk insurance contracts.\n\n\\begin{example}[Quantities of interest]\nIn this example, we let $M=D+5$, and defined the following quantities of interest. \n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{sexp:quantities-mitiadoption}For every $d\\in\\mathcal{D}:=\\{0,1,\\ldots,D\\}$, let $\\zeta^{(d+1)}_{t}(b,i,w)=\\INDI_{\\{\\widehat{d}_t(b,i)=d\\}}$. Then, we have\n$\\overline{\\zeta}^{(d+1)}_{t}=$ \\linebreak$\\PROB\\big[\\widehat{d}_t\\big(b^{\\pi^\\star}_{t-1},i^{\\pi^\\star}_{t-1}\\big)=d\\big]$, which corresponds to the probability that the self-mitigation measure $d$ is adopted in year $t$ under the decision policy $\\pi^\\star$.\n\\item Let $\\zeta^{(D+2)}_{t}(b,i,w)=e^{-tr}\\beta(\\widehat{d}_t(b,i))$. Then, $\\overline{Z}_{\\zeta^{(D+2)}}$ corresponds to the expected value of the discounted total self-mitigation expenses. \n\\item \\label{sexp:quantities-premium}Let $\\zeta^{(D+3)}_{t}(b,i,w)=e^{-tr}\\big(\\widehat{\\iota}_t(b,i)\\big(p^{\\mathcal{B}\\mathcal{M}}(b)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o}\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{n}\\mathrm{o},i\\ne\\mathrm{o}\\mathrm{n}\\}}\\big)+(1-\\widehat{\\iota}_t(b,i))\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n}\\}}\\big)$.\nThen, $\\overline{Z}_{\\zeta^{(D+3)}}$ corresponds to the expected value of the discounted total payment from the insured to the insurer. \n\\item \\label{sexp:quantities-mitieffect}Let $\\zeta^{(D+4)}_{t}\\big(b,i,w=(n,x_1,\\ldots,x_n)\\big)=e^{-tr}\\left(\\sum_{k=1}^{n}x_k-L(\\widehat{d}_t(b,i),w)\\right)$. Then, $\\overline{Z}_{\\zeta^{(D+4)}}$ corresponds to the expected value of the discounted total loss that is prevented by the self-mitigation measures. \n\\item \\label{sexp:quantities-benefit}Let $\\zeta^{(D+5)}_{t}(b,i,w)=e^{-tr}\\Big(\\widehat{j}_t(b,i,w)\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),w))\\Big)$, then $\\overline{Z}_{\\zeta^{(D+5)}}$ corresponds to the expected value of the discounted total insurance compensation the insured receives. \n\\end{enumerate}\n\\label{exp:quantities} \n\\end{example}\n\nAs a consequence of Theorem~\\ref{thm:dp}, let us now introduce a concrete algorithm based on the dynamic programming principle to solve the stochastic optimal control problem (\\ref{eqn:stocoptctr}), which is presented in Algorithm~\\ref{alg:dpconcrete}. \nIn addition to an optimal decision policy $\\pi^\\star$, Algorithm~\\ref{alg:dpconcrete} also computes the state transition kernels $\\big(P^\\star_t\\big)_{t=1:T}$ and the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ of the optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$, defined in (\\ref{eqn:dpalgo-kernel}) and (\\ref{eqn:dpalgo-marg}).\nMoreover, given $M$ quantities of interest $\\big(\\zeta^{(m)}(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ as input, Algorithm~\\ref{alg:dpconcrete} also computes their expected values defined in (\\ref{eqn:qoi-expval}) as well as their aggregate expected values defined in (\\ref{eqn:qoi-aggregate}). \nRemark~\\ref{rmk:dpalgo} provides detailed explanations of the inputs and outputs of Algorithm~\\ref{alg:dpconcrete} as well as explanations of some of the lines in Algorithm~\\ref{alg:dpconcrete}. \nTheorem~\\ref{thm:dpalgo} shows the correctness of Algorithm~\\ref{alg:dpconcrete}. \nMoreover, Remark~\\ref{rmk:comp} discusses the computational tractability of Algorithm~\\ref{alg:dpconcrete}.\n\n\n\n\n\n\\begin{algorithm}[p]\n\\KwIn{$\\mathcal{B}$, $\\mathcal{I}$, $\\mathcal{D}$, $\\mathcal{B}\\mathcal{M}$, $\\mathcal{B}\\mathcal{M}_{0}$, $p^{\\mathcal{B}\\mathcal{M}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}$, $\\beta(\\cdot)$, $\\gamma(\\cdot)$, $\\delta_{\\mathrm{i}\\mathrm{n}}(\\cdot)$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(\\cdot)$, $\\delta_{\\mathrm{r}\\mathrm{e}}$, $r$, $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$}\n\\KwOut{$\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$}\n\\nl $\\mathcal{V}_T(b,i)\\leftarrow0$ for all $b\\in\\mathcal{B},i\\in\\mathcal{I}$. \\label{alglin:dp-value-init} \\\\\n\\nl \\For{$t=T,T-1,\\ldots,1$}{\n\\nl \\label{alglin:dp-forloop-b}\\For{$b\\in\\mathcal{B}$}{\n\\nl $\\underline{b}\\leftarrow\\mathcal{B}\\mathcal{M}(b,0)$, $\\overline{b}\\leftarrow\\max\\{\\mathcal{B}\\mathcal{M}(b,c):c\\in\\R_+\\}$. \\COMMENT{$\\underline{b}$ and $\\overline{b}$ are the lowest and highest possible Bonus-Malus levels reachable from level $b$} \\label{alglin:dp-b-b-def} \\\\\n\\nl \\For{$\\underline{b}\\le b'\\le\\overline{b}$}{\n\\nl $\\alpha_t(b,b')\\leftarrow \\mathcal{V}_{t}(b',\\mathrm{o}\\mathrm{n})-\\mathcal{V}_{t}(\\underline{b},\\mathrm{o}\\mathrm{n})$. \\COMMENT{temporary value to simplify computation} \\label{alglin:dp-alpha-def} \\\\\n\\nl $\\mathcal{L}_t(b,b')\\leftarrow\\left\\{c\\in\\R_+:\\mathcal{B}\\mathcal{M}(b,c)=b',c>\\alpha_t(b,b')\\right\\}$. \\COMMENT{the set of potential insurance compensation amounts such that the optimal decision is to make a claim and the updated Bonus-Malus level will be $b'$; see also Remark~\\ref{rmk:dpalgo}}\\label{alglin:dp-L-def} \\\\\n}\n\\nl \\For{$i\\in\\mathcal{I}$}{\n\\nl \\label{alglin:dp-forloop-d}\\For{$d\\in\\mathcal{D}$}{\n\\nl $H_t(b,i,d,1)\\leftarrow \\mathcal{V}_t(\\underline{b},\\mathrm{o}\\mathrm{n})-\\sum_{\\underline{b}\\le b'\\le\\overline{b}}\\EXP\\bigg[\\INDI_{\\{\\mathcal{B}\\mathcal{M}(b,\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))=b'\\}}\\Big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha_t(b,b')\\Big)^+\\bigg]$.\\label{alglin:dp-H-def1} \\\\\n\\nl $H_t(b,i,d,0)\\leftarrow \\mathcal{V}_t\\big(\\mathcal{B}\\mathcal{M}_0(b,i)\\big)$. \\COMMENT{$H_t(b,i,d,\\iota)$ for $\\iota\\in\\{0,1\\}$ are temporary values to simplify the one-stage optimisation problem in Line~\\ref{alglin:dp-d-iota-def}; see also Remark~\\ref{rmk:dpalgo}}\\label{alglin:dp-H-def2} \\\\\n}\n\\nl $(\\widehat{d}_t(b,i),\\widehat{\\iota}_t(b,i))\\leftarrow\\argmin_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}} \\bigg\\{\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota)\\bigg\\}$.\\label{alglin:dp-d-iota-def} \\\\\n\\nl $\\widehat{j}_t(b,i,w)\\leftarrow\\INDI_{\\{\\widehat{\\iota}_t(b,i)=1\\}}\\INDI_{\\bigcup_{\\underline{b}\\le b'\\le\\overline{b}}\\mathcal{L}_t(b,b')}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),w))\\big)$.\\label{alglin:dp-j-def} \\\\\n\\nl $\\mathcal{V}_{t-1}(b,i)\\leftarrow e^{-r}\\min_{d\\in\\mathcal{D},\\,\\iota\\in\\{0,1\\}} \\bigg\\{\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota)\\bigg\\}$.\\label{alglin:dp-value-update} \\\\\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(b',i')\\big]\\leftarrow 0$ for all $(b',i')\\in\\mathcal{B}\\times\\mathcal{I}$. \\COMMENT{initialise all transition probabilities to 0} \\\\\n\\nl \\If{$\\widehat{\\iota}_t(b,i)=1$}{\n\\nl \\For{$\\underline{b}<b'\\le\\overline{b}$}{\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(b',\\mathrm{o}\\mathrm{n})\\big]\\leftarrow {\\PROB\\Big[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(\\widehat{d}_t(b,i),W))\\in\\mathcal{L}_t(b,b')\\Big]}$. \\COMMENT{compute the transition probability to level $b'$}\\label{alglin:dp-kern1} \\\\\n}\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow(\\underline{b},\\mathrm{o}\\mathrm{n})\\big]\\leftarrow 1-\\sum_{\\underline{b}<b'\\le\\overline{b}}P^\\star_t\\big[(b,i)\\rightarrow(b',\\mathrm{o}\\mathrm{n})\\big]$. \\label{alglin:dp-kern2} \\COMMENT{compute the transition probability to the lowest level}\\\\\n}\\nl \\Else{\n\\nl $P^\\star_t\\big[(b,i)\\rightarrow\\mathcal{B}\\mathcal{M}_0(b,i)\\big]\\leftarrow 1$. \\COMMENT{when the optimal decision is to not purchase insurance, the transition is deterministic} \\label{alglin:dp-kern3} \\\\\n}\n}\n}\n}\n\\nl $\\overline{P}_{0}^{\\star}(0,\\mathrm{n}\\mathrm{o})\\leftarrow 1$, $\\overline{P}_{0}^{\\star}(b,i)\\leftarrow 0$ for all $(b,i)\\ne(0,\\mathrm{n}\\mathrm{o})$. \\label{alglin:dp-marg-init} \\\\\n\\nl \\For{$t=1,2,\\ldots,T$}{\n\\nl For all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $\\overline{P}_{t}^{\\star}(b,i)\\leftarrow\\sum_{(b',i')\\in\\mathcal{B}\\times\\mathcal{I}}P^\\star_t[(b',i')\\rightarrow(b,i)]\\overline{P}_{t-1}^{\\star}(b',i')$. \\label{alglin:dp-marg-iter} \\\\\n\\nl For $m=1,\\ldots,M$, $\\overline{\\zeta}^{(m)}_t\\leftarrow\\sum_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}\\EXP[\\zeta^{(m)}_t(b,i,W)]\\overline{P}_{t-1}^{\\star}(b,i)$. \\label{alglin:dp-qoi} \\\\\n}\n\\nl For $m=1,\\ldots,M$, $\\overline{Z}_{\\zeta^{(m)}}\\leftarrow\\sum_{t=1}^T\\overline{\\zeta}_t^{(m)}$. \\label{alglin:dp-qoi-aggregate} \\\\\n\\nl Define $\\pi^\\star=(d^\\star_t,\\iota^\\star_t,j^\\star_t)_{t=1:T}$ by (\\ref{eqn:dpconcept-opt-process}).\\\\\n\\nl \\Return $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$. \n    \\caption{{\\bf Dynamic Programming for Optimal Cybersecurity Provisioning}}\n    \\label{alg:dpconcrete}\n\\end{algorithm}\n\n\\begin{remark}[Details about Algorithm~\\ref{alg:dpconcrete}]\nThe inputs of Algorithm~\\ref{alg:dpconcrete} are explained as follows.\n\\begin{itemize}\n\\item $\\mathcal{B}$, $\\mathcal{I}$, $\\mathcal{D}$, $\\mathcal{B}\\mathcal{M}$, $\\mathcal{B}\\mathcal{M}_{0}$, $p^{\\mathcal{B}\\mathcal{M}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}$, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}$, $\\beta(\\cdot)$, $\\gamma(\\cdot)$, $\\delta_{\\mathrm{i}\\mathrm{n}}(\\cdot)$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(\\cdot)$, $\\delta_{\\mathrm{r}\\mathrm{e}}$ are specified in Section~\\ref{ssec:mitigation}, Section~\\ref{ssec:cyberinsurance}, and Section~\\ref{ssec:bonusmalus}.\n\\item $r\\ge0$ specifies the discount factor $e^{-r}$ in the objective of the optimal cybersecurity provisioning process.\n\\item $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ specify $M$ quantities of interest that satisfy (\\ref{eqn:qoi}) whose expected values (\\ref{eqn:qoi-expval}) and aggregate expected values (\\ref{eqn:qoi-aggregate}) will be computed in Algorithm~\\ref{alg:dpconcrete}. Some examples of quantities of interest are shown in Example~\\ref{exp:quantities}. \n\\end{itemize}\nThe outputs of Algorithm~\\ref{alg:dpconcrete} are explained as follows.\n\\begin{itemize}\n\\item $\\pi^{\\star}$ is the optimal decision policy computed by the dynamic programming algorithm and $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$ is equal to the optimal value $\\mathbb{V}_0$ of the stochastic optimal control problem. The optimality of $\\pi^\\star$ is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \n\\item $\\big(P^\\star_t\\big)_{t=1:T}$ are the transition kernels of the discrete-time time-inhomogeneous Markov chain \\linebreak$\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ that are defined in (\\ref{eqn:dpalgo-kernel}). $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ are the marginal state occupancy probabilities of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}). The correctness of these outputs is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo2}. \n\\item $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$ are the expected values of the quantities of interest $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ defined in (\\ref{eqn:qoi-expval}). $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ are the aggregate expected values of these quantities of interest defined in (\\ref{eqn:qoi-aggregate}). The correctness of these outputs is shown in Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo3}. \n\\end{itemize}\nBelow is a list explaining some of the lines in Algorithm~\\ref{alg:dpconcrete}. \n\\begin{itemize}\n\\item Line~\\ref{alglin:dp-b-b-def} computes the lowest possible Bonus-Malus level $\\underline{b}$ and the highest possible Bonus-Malus level $\\overline{b}$ that the insured can transition to from level $b$. We only iterate through the Bonus-Malus levels between $\\underline{b}$ and $\\overline{b}$ in the remainder of the for-loop starting from Line~\\ref{alglin:dp-forloop-b}.\n\\item Line~\\ref{alglin:dp-L-def} computes a set $\\mathcal{L}_t(b,b')\\subseteq\\R_+$ which contains all potential insurance compensation amounts (i.e., $\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))$) such that the optimal decision of the insured is to make a claim, and that the updated Bonus-Malus level after making the claim is $b'$. \nIn particular, the following claim holds true.\n\\begin{enumerate}[label=\\underline{Claim~\\arabic*}:,leftmargin=3.8em]\n\\item for all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $d\\in\\mathcal{D}$, the following equivalence holds:\n\\begin{align*}\n\\begin{split}\ng_{t}(b,i,d,1,1,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,1,1,w)\\big)&< g_{t}(b,i,d,1,0,w)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,1,0,w)\\big)\\\\\n&\\Updownarrow\\\\\n\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,w))&\\in\\bigcup_{\\underline{b}\\le b'\\le\\overline{b}}\\mathcal{L}_t(b,b').\n\\end{split}\n\\end{align*}\n\\end{enumerate}\nThis claim is justified in (\\ref{eqn:dpalgo-claim1-proof1}) and (\\ref{eqn:dpalgo-claim1-proof2}) in the proof of Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \nMoreover, notice that $\\mathcal{L}_t(b,b')$ is an interval since the set $\\left\\{c\\in\\R_+:\\mathcal{B}\\mathcal{M}(b,c)=b'\\right\\}$ is an interval in $\\R_+$ by the assumption that the function $\\mathcal{B}\\mathcal{M}(b,\\cdot)$ is non-decreasing.\n\\item In the for-loop starting from Line~\\ref{alglin:dp-forloop-d}, we compute temporary values $(H_t(b,i,d,\\iota))_{d\\in\\mathcal{D},\\iota\\in\\{0,1\\}}$ which are subsequently used in Line~\\ref{alglin:dp-d-iota-def} and Line~\\ref{alglin:dp-value-update}. In fact, the following claim holds true.\n\\begin{enumerate}[label=\\underline{Claim~\\arabic*}:,leftmargin=3.8em]\n\\setcounter{enumi}{1}\n\\item for all $(b,i)\\in\\mathcal{B}\\times\\mathcal{I}$, $d\\in\\mathcal{D}$, $\\iota\\in\\{0,1\\}$, it holds that\n\\begin{align*}\n\\begin{split}\n&\\phantom{=}\\;\\;\\EXP\\bigg[\\min_{j\\in\\{0,1\\}}\\Big\\{g_{t}(b,i,d,\\iota,j,W)+\\mathcal{V}_{t}\\big(f_{t}(b,i,d,\\iota,j,W)\\big)\\Big\\}\\bigg]\\\\\n&=\\beta(d)+\\iota p^{\\mathcal{B}\\mathcal{M}}(b,t)+\\delta_{\\mathrm{i}\\mathrm{n}}(t)\\INDI_{\\{i=\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)\\INDI_{\\{i=\\mathrm{o}\\mathrm{n},\\iota=0\\}}\\\\\n&\\qquad+\\delta_{\\mathrm{r}\\mathrm{e}}\\INDI_{\\{i\\ne\\mathrm{o}\\mathrm{n},i\\ne\\mathrm{n}\\mathrm{o},\\iota=1\\}}+\\EXP\\big[L(d,W)\\big]+H_t(b,i,d,\\iota).\n\\end{split}\n\\end{align*}\n\\end{enumerate}\nThis claim is justified in (\\ref{eqn:dpalgo-claim2-proof1}) and (\\ref{eqn:dpalgo-claim2-proof2}) in the proof of Theorem~\\ref{thm:dpalgo}\\ref{sthm:dpalgo1}. \n\\item The main step of the dynamic programming algorithm are in Line~\\ref{alglin:dp-value-init}, Line~\\ref{alglin:dp-d-iota-def}, Line~\\ref{alglin:dp-j-def}, and Line~\\ref{alglin:dp-value-update}. In Line~\\ref{alglin:dp-value-init}, the values of $(\\mathcal{V}_T(b,i))_{(b,i)\\in\\mathcal{B}\\times\\mathcal{I}}$ are all initialised to 0 according to (\\ref{eqn:dpvalue}). \nIn Line~\\ref{alglin:dp-d-iota-def} and Line~\\ref{alglin:dp-j-def}, the one-stage optimisers $(\\widehat{d}_t,\\widehat{\\iota}_t,\\widehat{j}_t)$ are computed as in (\\ref{eqn:dpconcept-d-iota}) and (\\ref{eqn:dpconcept-j}), due to \\underline{Claim~1} and \\underline{Claim~2} above. \nIn Line~\\ref{alglin:dp-value-update}, the value of $\\mathcal{V}_t(b,i)$ is computed as in (\\ref{eqn:dpvalue}), due to \\underline{Claim~2} above.\n\\end{itemize}\n\\label{rmk:dpalgo}\n\\end{remark}\n\n\\begin{theorem}\nLet us assume that all inputs of Algorithm~\\ref{alg:dpconcrete} are set according to Remark~\\ref{rmk:dpalgo}, and let $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})$, $\\pi^{\\star}$, $\\big(P^\\star_t\\big)_{t=1:T}$, $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$, $\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$, and $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ be the outputs of Algorithm~\\ref{alg:dpconcrete}. \nThen, the following statements hold.\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\label{sthm:dpalgo1}$\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=\\mathbb{V}_0$ and $\\pi^\\star$ is an optimal decision policy for (\\ref{eqn:stocoptctr}), i.e., $\\mathcal{V}_0(0,\\mathrm{n}\\mathrm{o})=V^{\\pi^\\star}_0=\\mathbb{V}_0$. \n\n\\item \\label{sthm:dpalgo2}$\\big(P^\\star_t\\big)_{t=1:T}$ are equal to the transition kernels of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-kernel}), and $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ are equal to the marginal state occupancy probabilities of $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ defined in (\\ref{eqn:dpalgo-marg}).\n\n\\item \\label{sthm:dpalgo3}$\\big(\\overline{\\zeta}_{t}^{(m)}\\big)_{t=1:T,m=1:M}$ are equal to the expected values of the quantities of interest $\\big(\\zeta^{(m)}_t(\\cdot,\\cdot,\\cdot)\\big)_{m=1:M}$ defined in (\\ref{eqn:qoi-expval}), and $\\big(\\overline{Z}_{\\zeta^{(m)}}\\big)_{m=1:M}$ are equal to the aggregate expected values of these quantities of interest defined in (\\ref{eqn:qoi-aggregate}).\n\n\\end{enumerate}\n\\label{thm:dpalgo}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\n\\begin{remark}\nAssume that the following quantities either admit an analytically tractable expression, or can be efficiently approximated to high numerical precision:\n\\begin{enumerate}[label=(\\roman*)]\n\\item the expectation $\\EXP\\big[L(d,W)\\big]=\\EXP[N]\\EXP\\big[(X-d)^+\\big]$, where $d\\in\\mathcal{D}$; \\label{srmk:comp1}\n\\item the expectation $\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\Big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\Big)^+\\right]$, where $b\\in\\mathcal{B}$, $d\\in\\mathcal{D}$, $t\\in\\{1,\\ldots,T\\}$, $\\alpha\\ge0$, and $I\\subset\\R_+$ is an interval; \\label{srmk:comp2}\n\\item the probability $\\PROB\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]$, where $b\\in\\mathcal{B}$, $d\\in\\mathcal{D}$, $t\\in\\{1,\\ldots,T\\}$, and $I\\subset\\R_+$ is an interval; \\label{srmk:comp3}\n\\item the expectation $\\EXP\\big[\\zeta^{(m)}_t(b,i,W)\\big]$, where $b\\in\\mathcal{B}$, $i\\in\\mathcal{I}$, $m\\in\\{1,\\ldots,M\\}$, $t\\in\\{1,\\ldots,T\\}$. \\label{srmk:comp4}\n\\end{enumerate}\nThen, Algorithm~\\ref{alg:dpconcrete} is computationally tractable, meaning that quantities in Algorithm~\\ref{alg:dpconcrete} can either be computed exactly or efficiently approximated to high numerical precision. \nIn addition, since $\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))$ is bounded above by $l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)$, we can assume without loss of generality that the interval $I$ in \\ref{srmk:comp2} and \\ref{srmk:comp3} above is bounded. A concrete model in which Algorithm~\\ref{alg:dpconcrete} is computationally tractable will be introduced in Section~\\ref{sec:g-and-h}. \n\\label{rmk:comp}\n\\end{remark}\n\n\n\n\\subsection{Discussion About the Pricing of the Insurance Premium}\n\\label{ssec:pricing}\nOne important consideration of the cyber risk insurer is the choice of the annual premium. Conventionally, for a fixed self-mitigation measure $d\\in\\mathcal{D}$, a fixed deductible $l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}\\ge 0$, and a fixed maximum compensation $l_{\\max}\\ge0$, one may consider the risk premium $\\EXP[(L(d,W)-l_{\\mathrm{d}\\mathrm{t}\\mathrm{b}})^+\\wedge l_{\\max}]$. \nHowever, these quantities are variable in our dynamic model with the Bonus-Malus system.\nMoreover, the insured may choose to withdraw from the cyber risk insurance contract, thus further complicating the matter.\n\nEven though the optimal cybersecurity provisioning process minimises the total cybersecurity cost from the insured's point of view, the insurer can use the quantities of interest computed from this process to evaluate the insurance contract and compare different contract design options.\nOne of the evaluation criteria is how high the insurer can price the premium while retaining their customers, that is, what is the highest premium such that the cyber risk insurance policy remains attractive to a rational insured. \nTo answer this question, one can analyse the relationship between the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ and the premium.\nWhen the premium is high, the insured may opt not to purchase the cyber risk insurance policy, or may withdraw from the insurance contract prematurely due to a transition into a higher Bonus-Malus level. \nThis evaluation criterion is crucial for the insurer, since a high retention rate would allow the insurer to build a large pool of customers with similar risk exposures.\n\nAnother criterion is the range of premium under which a rational insured would choose to adopt a self-mitigation measure. \nThe reasoning here is that if the insurance premium is set too low, then the insured will see cyber risk insurance as a low-cost alternative to self-mitigation measures and will thus choose not to adopt any self-mitigation measure.  \nThis range can be determined by studying the relationship between the values of\n$\\big(\\overline{\\zeta}^{(d+1)}_t\\big)_{t=1:T,d\\in\\mathcal{D}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-mitiadoption} and the premium.\nMoreover, the value of $\\overline{Z}_{\\zeta^{(D+4)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-mitieffect} reflects the discounted total amount of loss prevented by the adoption of self-mitigation measures, and its relationship with the premium can also be studied.\nThese are important criteria due to the positive externalities generated by organisations adopting self-mitigation measures against cyber threats.\nCombining the criterion of loss prevention with the criterion of insurance customer retention, one can study whether the adoption of cyber risk insurance disincentivises the adoption of self-mitigation measures, which is an issue known as moral hazard.\n\n\nAn important question related to the design of the insurance contract is how much the insurer can financially gain from offering the cyber risk insurance policy. One simple criterion is to look at the discounted expectation of the difference between the total payment from the insured to the insurer and the total insurance compensation. Specifically, let $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}\\equiv\\overline{Z}_{\\zeta^{(D+3)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-premium} and let $\\overline{Z}_{\\mathrm{c}\\mathrm{p}}\\equiv\\overline{Z}_{\\zeta^{(D+5)}}$ in Example~\\ref{exp:quantities}\\ref{sexp:quantities-benefit}. \nThen, one can see $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ as the insurer's expected profit from the cyber risk insurance policy.\nHowever, this approach implicitly makes the zero-sum assumption, i.e., the insurer's expected profit is the negative of the expected net payment from the insured to the insurer. \nSince we assume that the insured always has the option to not purchase cyber risk insurance, which would result in zero net payment to the insurer, the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ will always be non-positive. \nIn particular, in the case where the insurance policy is not purchased, the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ will be exactly 0.\nIn reality, there are a few important practical considerations that this zero-sum approach neglects.\nFirst, the insured and the insurer often perceive cyber risk differently. \nSpecifically, this is due to the insured having impartial or incomplete information as to the frequency and severity of the risk they face relative to an insurer who is privy to a portfolio of insured, giving them a competitive advantage in their assessments of such loss processes.\nSecond, the zero-sum approach assumes that the insured will act optimally according to the optimal cybersecurity provisioning process.\nIn reality, the insured will typically act sub-optimally due to the practical complexity of the required computation being beyond the scope and the available resources of the decision makers \\citep{targino2013optimal}.\nThird, regulatory transparency requirements will also play an important role in determining the profit margins that may arise if such products are issued.\nThe lack of these considerations indicates some of the limitations of this zero-sum approach. \nNevertheless, the relationship between the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ and the premium does shed some light on an important aspect of cyber risk insurance contract, that is, whether it is possible to make this quantity arbitrarily close to 0 by setting the premium appropriately. \nIf an insurance contract design allows such possibility, then there is a way to set the premium such that the insured becomes indifferent between purchasing and not purchasing the insurance policy up to an arbitrarily small tolerance, at which point the value of $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ would be approximately maximised. \nThen, the real profit of the insurer should become more preferable than the case where the value of $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ is far from zero.\nOn the contrary, if a design does not allow such possibility, then there is no way to set the premium such that the insured becomes approximately indifferent between purchasing and not purchasing the insurance policy, indicating that there is space for improving the contract design in order to hopefully further improve the real profit of the insurer.\nWe will discuss the aforementioned important criteria when evaluating the design of cyber risk insurance contracts with a concrete example in Section~\\ref{sec:exp}.\n\n\n\\section{Modelling Cyber Loss with Truncated g-and-h Distribution}\n\\label{sec:g-and-h}\n\nIn this section, we adopt specific distributional assumptions about the random variable $X$, which corresponds to the severity of a single cyber loss event. \nStudies such as \\cite{maillart2010heavy} and \\cite{wheatley2016extreme} have shown that the severity of cyber loss events have heavy-tailed distributions. \nIt is well-known that the g-and-h family introduced by \\cite{tukey1977exploratory} contains distributions with a wide range of skewness and kurtosis (see, e.g., Figure~3 of \\cite{dutta2006tale}), which makes it suitable for modelling Operational Risk \\citep{dutta2006tale,peters2006bayesian,cruz2015fundamental,peters2015advances}. \nIn \\cite{dutta2006tale}, the following advantages of the g-and-h distribution are discussed:\n\\begin{itemize}\n\\item it is flexible and it fits well to real data under many different circumstances, e.g., when considering losses of a company as a whole and when considering individual business lines or event types;\n\\item it produces realistic estimations of the Operational Risk capital;\n\\item it is easy to simulate random samples from.\n\\end{itemize}\nThe parameters in the g-and-h distribution can be robustly estimated based on quantiles~\\citep{xu2014robust} or L-moments~\\citep{peters2016estimating}. \nDue to these properties, we adopt the g-and-h distribution as a particular model for the severity of cyber loss events in this section. \n\nThe g-and-h distribution is a four-parameter family of distributions, given by the following definition:\n\\begin{align}\n\\begin{split}\n\\widetilde{X}\\text{ follows a g-and-h}&(\\alpha,\\varsigma,g,h)\\text{ distribution, if}\\\\\n\\widetilde{X}&=\\alpha+\\varsigma Y_{g,h}(Z),\\\\\n\\text{where }Z&\\sim\\text{Normal}(0,1),\\\\\nY_{g,h}(z)&:=\\begin{cases}\n\\frac{\\exp(gz)-1}{g}\\exp\\left(\\frac{hz^2}{2}\\right) & \\text{if }g\\ne 0,\\\\\nz\\exp\\left(\\frac{hz^2}{2}\\right) & \\text{if }g=0,\n\\end{cases}\n\\end{split}\n\\label{eqn:g-and-hdef}\n\\end{align}\nwhere $\\alpha\\in\\R$ is the location parameter, $\\varsigma>0$ is the scale parameter, $g\\in\\R$ is the skewness parameter, and $h\\ge0$ is the kurtosis parameter. \nIn this paper, we assume that the parameters $\\alpha$, $\\varsigma$, $g$, and $h$ are fixed and known.  \nBy (\\ref{eqn:g-and-hdef}), the distribution function of $\\widetilde{X}$ is given by\n\\begin{align}\n\\begin{split}\nF_{\\widetilde{X}}(x):=\\PROB[\\widetilde{X}\\le x]=\\Phi\\left(Y_{g,h}^{-1}\\left(\\tfrac{x-\\alpha}{\\varsigma}\\right)\\right),\n\\end{split}\n\\end{align}\nwhere $Y_{g,h}^{-1}$ denotes the inverse function of $Y_{g,h}$, and $\\Phi$ denotes the distribution function of the standard normal distribution. \nEven though $Y_{g,h}^{-1}$ cannot be expressed analytically, it can be efficiently evaluated using a standard root-finding procedure such as the bisection method and the Newton's method. Therefore, we treat $Y_{g,h}^{-1}$ as a tractable function. \nThe g-and-h distribution has the property that the $m$-th moment of $\\widetilde{X}$ exists when $h<\\frac{1}{m}$ (see, e.g., Appendix~D of \\cite{dutta2006tale}). Since we consider losses that are positively skewed and have finite expectation, from now on, we assume that $g>0$ and $0\\le h<1$. \n\nSince cyber losses are positive, we introduce a truncated version of the g-and-h distribution. \n\\begin{definition}[Truncated g-and-h distribution] For $\\alpha\\in\\R,\\varsigma>0,g>0,h\\in[0,1)$, the random variable $X$ has truncated g-and-h distribution with parameters $\\alpha,\\varsigma,g,h$, denoted by $X\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$, if $X$ has distribution function \n\\begin{align}\nF_X(x):=\\PROB[X\\le x]=\\PROB[\\widetilde{X}\\le x|\\widetilde{X}>0],\n\\label{eqn:tr-g-and-hdef}\n\\end{align}\nwhere $\\widetilde{X}\\sim\\text{g-and-h}(\\alpha,\\varsigma,g,h)$. \n\\label{def:tr-g-and-h}\n\\end{definition}\n\nThe next lemma shows some useful properties of the truncated g-and-h distribution.\n\\begin{lemma}\nSuppose that $X\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$ for $\\alpha\\in\\R,\\varsigma>0,g>0,h\\in[0,1)$. Then, the following statements hold.\n\\begin{enumerate}[label=(\\roman*)]\n\\item The distribution function of $X$ is given by\n\\begin{align}\n\\begin{split}\nF_{X}(x)=\\begin{cases}\n\\frac{F_{\\widetilde{X}}(x)-F_{\\widetilde{X}}(0)}{1-F_{\\widetilde{X}}(0)} & \\text{if }x>0,\\\\\n0 & \\text{if }x\\le0,\n\\end{cases}\n\\end{split}\n\\label{eqn:tr-g-and-h-df}\n\\end{align}\nwhere $F_{\\widetilde{X}}$ is defined in (\\ref{eqn:g-and-hdef}). \n\\label{slem:tr-g-and-h1}\n\\item Suppose that $U\\sim\\text{Uniform}[0,1]$, and let\n\\begin{align}\nX_U:=\\alpha+\\varsigma Y_{g,h}\\Big(\\Phi^{-1}\\Big(U+(1-U)F_{\\widetilde{X}}(0)\\Big)\\Big),\n\\label{eqn:tr-g-and-h-inv}\n\\end{align}\nthen $X_U\\sim\\text{Tr-g-and-h}(\\alpha,\\varsigma,g,h)$. \n\\label{slem:tr-g-and-h2}\n\\item For any $\\gamma\\ge0$, the expectation $\\EXP\\big[(X-\\gamma)^+\\big]$ is given by:\n\\begin{align}\n\\begin{split}\n\\EXP\\big[(X-\\gamma)^+\\big]&=\\frac{\\varsigma}{(1-F_{\\widetilde{X}}(0))g\\sqrt{1-h}}\\Bigg[\\exp\\left(\\frac{g^2}{2(1-h)}\\right)\\Phi\\left(\\left(\\frac{g}{1-h}-Y_{g,h}^{-1}\\left(\\tfrac{\\gamma-\\alpha}{\\varsigma}\\right)\\right)\\sqrt{1-h}\\right)\\\\\n&\\qquad-\\Phi\\left(-Y_{g,h}^{-1}\\left(\\tfrac{\\gamma-\\alpha}{\\varsigma}\\right)\\sqrt{1-h}\\right)\\Bigg]+\\frac{(\\alpha-\\gamma)(1-F_{\\widetilde{X}}(\\gamma))}{1-F_{\\widetilde{X}}(0)}.\n\\end{split}\n\\label{eqn:tr-g-and-h-mom}\n\\end{align}\n\\label{slem:tr-g-and-h3}\n\\end{enumerate}\n\\label{lem:tr-g-and-h}\n\\end{lemma}\n\\begin{proof}\nSee Appendix~\\ref{apx:proofs}. \n\\end{proof}\n\nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h2} allows us to efficiently generate random samples from the severity distribution $F_X$, thus allowing us to approximate the distribution of quantities of interest in Example~\\ref{exp:quantities} via Monte Carlo.\nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h3} shows that the assumption~\\ref{srmk:comp1} in Remark~\\ref{rmk:comp} is satisfied as long as the expected value of the frequency distribution, i.e., $\\EXP[N]$, is also tractable. \nLemma~\\ref{lem:tr-g-and-h}\\ref{slem:tr-g-and-h1} provides the distribution function of $X$ that can be used to approximate the distribution function of $L(d,W)$. Concretely, by adopting the fast Fourier transform (FFT) approach with exponential tilting (see, e.g., \\cite{embrechts2009panjer,cruz2015fundamental}), we approximate the distribution function of $L(d,W)$, denoted by $F_{L(d,W)}$, by a finitely supported discrete distribution $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\big\\{a^{(d)}_j\\le x\\big\\}}$, where $\\big(a^{(d)}_j\\big)_{j\\in\\mathcal{A}}\\subset\\R_+$ is a finite set of atoms and $\\big(p^{(d)}_j\\big)_{j\\in\\mathcal{A}}$ are the corresponding probabilities. \nThe details of the FFT approach with exponential tilting are shown in Algorithm~\\ref{alg:fft}. \nAfter obtaining $(\\widehat{F}_{L(d,W)})_{d\\in\\mathcal{D}}$ from Algorithm~\\ref{alg:fft}, the quantities $\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\big)^+\\!\\right]$ and $\\PROB\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]$ in Remark~\\ref{rmk:comp} can be approximated by finite sums:\n\\begin{align*}\n\\EXP\\!\\left[\\INDI_{I}(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W)))\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))-\\alpha\\big)^+\\!\\right]&\\approx\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\!\\INDI_{I}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)\\big)\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)-\\alpha\\big)^+,\\\\\n\\PROB\\!\\left[\\lambda^{\\mathcal{B}\\mathcal{M}}(b,t,L(d,W))\\in I\\right]&\\approx\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\!\\INDI_{I}\\big(\\lambda^{\\mathcal{B}\\mathcal{M}}\\big(b,t,a^{(d)}_j\\big)\\big).\n\\end{align*}\nOne may increase the granularity parameter $K_{\\mathrm{g}\\mathrm{r}}$ in Algorithm~\\ref{alg:fft} to increase the precision of numerical approximation. \nConsequently, assumptions~\\ref{srmk:comp2} and \\ref{srmk:comp3} in Remark~\\ref{rmk:comp} are satisfied, and hence, Algorithm~\\ref{alg:dpconcrete} is tractable and efficient in this setting. In particular, Algorithm~\\ref{alg:fft} only needs to be executed once before executing Algorithm~\\ref{alg:dpconcrete}. \n\n\\begin{algorithm}[t]\n\\KwIn{$\\mathcal{D}$, $F_X(\\cdot)$, $\\psi_N(\\cdot)$, $\\gamma(\\cdot)$, $\\overline{l}$, $K_{\\mathrm{g}\\mathrm{r}}\\in\\N$, $\\theta>0$}\n\\KwOut{$(a^{(d)}_j,p_j^{(d)})_{j\\in\\mathcal{A},d\\in\\mathcal{D}}$, $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\{a^{(d)}_j\\le x\\}}$ for each $d\\in\\mathcal{D}$}\n\\nl $\\varepsilon\\leftarrow(2^{K_{\\mathrm{g}\\mathrm{r}}}-1)^{-1}\\overline{l}$, $\\mathcal{A}\\leftarrow\\{0,1,\\ldots,2^{K_{\\mathrm{g}\\mathrm{r}}}-1\\}$. \\\\\n\\nl \\For{$d\\in\\mathcal{D}$}{\n\\nl $a^{(d)}_j\\leftarrow j\\epsilon$ for each $j\\in\\mathcal{A}$. \\\\\n\\nl $f^{(d)}_{j}\\leftarrow \\exp(-j\\theta)\\left[F_{X,d}(j\\varepsilon+\\frac{1}{2}\\varepsilon)-F_{X,d}(j\\varepsilon-\\frac{1}{2}\\varepsilon)\\right]$ for each $j\\in\\mathcal{A}$, where $F_{X,d}(y):=F_X\\big(y+\\gamma(d)\\big)\\INDI_{\\{y\\ge 0\\}}$. \\\\\n\\nl $\\varphi^{(d)}_j\\leftarrow\\sum_{k\\in\\mathcal{A}}\\exp(i\\pi2^{1-K_{\\mathrm{g}\\mathrm{r}}} jk)f^{(d)}_k$ for each $j\\in\\mathcal{A}$ via the FFT algorithm. \\\\\n\\nl $\\psi^{(d)}_j\\leftarrow\\psi_N(\\varphi^{(d)}_j)$ for each $j\\in\\mathcal{A}$. \\\\\n\\nl $p^{(d)}_j\\leftarrow\\exp(j\\theta)2^{-K_{\\mathrm{g}\\mathrm{r}}}\\sum_{k\\in\\mathcal{A}}\\exp(-i\\pi2^{1-K_{\\mathrm{g}\\mathrm{r}}} jk)\\psi^{(d)}_k$ for each $j\\in\\mathcal{A}$ via the inverse FFT algorithm. \\\\\n}\n\\nl \\Return $(a^{(d)}_j,p_j^{(d)})_{j\\in\\mathcal{A},d\\in\\mathcal{D}}$, $\\widehat{F}_{L(d,W)}(x)=\\sum_{j\\in\\mathcal{A}}p^{(d)}_j\\INDI_{\\{a^{(d)}_j\\le x\\}}$ for each $d\\in\\mathcal{D}$.\\\\\n    \\caption{{\\bf Fast Fourier Transform Approach with Exponential Tilting for Approximating $F_{L(d,W)}$} (see \\cite{embrechts2009panjer})}\n    \\label{alg:fft}\n\\end{algorithm}\n\n\\section{Numerical Experiments}\n\\label{sec:exp}\n\nIn Section~\\ref{sec:stocoptctr} and Section~\\ref{sec:g-and-h}, we formulated the optimal cybersecurity provisioning problem as a finite horizon stochastic optimal control problem, and developed a dynamic programming algorithm, i.e., Algorithm~\\ref{alg:dpconcrete}, to efficiently solve the problem under the assumption that the loss severity follows the truncated g-and-h distribution. Algorithm~\\ref{alg:dpconcrete} not only computes the optimal cybersecurity provisioning policy $\\pi^\\star$ for the insured, but also computes related quantities of interest, including the marginal state occupancy probabilities $\\big(\\overline{P}^\\star_t\\big)_{t=0:T}$ of the optimally controlled process $\\big(b^{\\pi^\\star}_t,i^{\\pi^\\star}_t\\big)_{t=0:T}$ as well as other quantities of interest such as those illustrated in Example~\\ref{exp:quantities}.\nAs discussed in Section~\\ref{ssec:pricing}, these quantities can guide the insurer when designing a suitable cyber risk insurance contract with a Bonus-Malus system. \nIn this section, we demonstrate how Algorithm~\\ref{alg:dpconcrete} aids the insurer when designing a cyber risk insurance contract and the benefits of the Bonus-Malus system by a numerical experiment.\\footnote{The code used in this work for the experiment is available on GitHub: \\url{https:\/\/github.com\/qikunxiang\/CyberInsuranceBonusMalus}} \nIn particular, we investigate two aspects of the cyber risk insurance contract with Bonus-Malus discussed in Section~\\ref{ssec:pricing}. \nThe first aspect is the issue of moral hazard, that is, whether the presence of the cyber risk insurance contract disincentivises the adoption of self-mitigation measures. \nThe second aspect is whether the Bonus-Malus system provides benefits to the insurer in terms of increased customer retention rates and expected profits.\n\n\\subsection{Experimental Settings}\n\nWe assume that all monetary quantities, including the severity of cyber loss events, the insurance premium, and the annual investment required by the self-mitigation measures are adjusted to the scale of the insured (e.g., their average annual revenue) and are unit-free. \nWe consider insurance policies that last for 20 years, that is, $T=20$. The discount factor $e^{-r}$ is fixed at 0.95. \nIn the cyber loss model, we let the frequency distribution be the Poisson distribution with rate 0.8. We set the severity distribution to be $\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$, where the $g$ and $h$ parameters are set to be similar to those estimated in \\cite{dutta2006tale} from real Operational Risk data (see Table~8 of \\cite{dutta2006tale}). \nWe would like to remark that the heaviness of the tail of the loss severity distribution (i.e., the parameter $h$ in the truncated $g$-and-$h$ distribution) determines the probability of extreme risk events and is crucial in the computation of capital estimate \\citep{dutta2006tale}. Therefore, it is important that we specify a realistic value of the parameter $h$.\nIn Algorithm~\\ref{alg:fft}, we fix $\\overline{l}=10000$, $K_{\\mathrm{g}\\mathrm{r}}=20$, $\\theta=\\frac{20}{2^{K_{\\mathrm{g}\\mathrm{r}}}}=3.0518\\times10^{-4}$. \n\nFor simplicity, we consider the situation where only a single self-mitigation measure is available, that is, $D=1$. This self-mitigation measure requires an annual investment of 0.5, and has the effect of preventing 70\\% of the incoming cyber loss events and decreasing the severity of the remaining events by the 70th percentile of the severity distribution, that is, $\\beta(1)=0.5$, $\\gamma(1)=F^{-1}_X(0.7)$, where $X\\sim\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$. \nWe consider the following simple cyber risk insurance policy with Bonus-Malus system. Let $\\mathcal{B}=\\{-2,-1,0,1\\}$, and let the functions $\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$ and $\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$ be specified in Table~\\ref{tab:bm} below.\n\n\\begin{table}[h]\n\\label{tab:bm}\n\\caption{The $\\mathcal{B}\\mathcal{M}(\\cdot,\\cdot)$ and $\\mathcal{B}\\mathcal{M}_0(\\cdot,\\cdot)$ functions that represent the Bonus-Malus update rules}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n\\multicolumn{2}{|c|}{\\multirow{2}{*}{$\\mathcal{B}\\mathcal{M}(b_{t-1},C_t)$}} & \\multicolumn{2}{|c|}{$C_t$} \\\\ \n\\cline{3-4}\n\\multicolumn{2}{|c|}{} & $=0$ & $>0$ \\\\ \n\\cline{1-4}\n\\multirow{4}{*}{$b_{t-1}$} & $-2$ & $-2$ & $1$ \\\\ \n\\cline{2-4}\n & $-1$ & $-2$ & $1$ \\\\ \n\\cline{2-4}\n & $0$ & $-1$ & $1$ \\\\ \n\\cline{2-4}\n & $1$ & $0$ & $1$ \\\\ \n\\hline \n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n\\multicolumn{2}{|c|}{\\multirow{2}{*}{$\\mathcal{B}\\mathcal{M}_0(b_{t-1},i_{t-1})$}} & \\multicolumn{2}{|c|}{$i_{t-1}$} \\\\ \n\\cline{3-4}\n\\multicolumn{2}{|c|}{} & $\\text{on}$ & $\\text{off}_1$ \\\\ \n\\cline{1-4}\n\\multirow{4}{*}{$b_{t-1}$} & $-2$ & $(-2,\\text{off}_1)$ & $(-1,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $-1$ & $(-1,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $0$ & $(0,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\cline{2-4}\n & $1$ & $(1,\\text{off}_1)$ & $(0,\\text{off}_1)$ \\\\ \n\\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\nThe above settings mean that when the contract is activated, the insured is migrated to level 1 in the subsequent policy year whenever a claim is made. When the insured does not make any claim in a policy year, their policy is migrated downwards by one level in the subsequent policy years until it reaches level $-2$. When the contract is deactivated, if the insured's policy is in level 1, it is migrated back to level 0 after one year. Otherwise, the policy is migrated upwards by one level each year until it reaches level 0. \nIn the experiment, we let the base premium $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$ be an adjustable parameter that is varied between 0 and 7 with an increment of $0.005$, and set the premium to be $60\\%,80\\%,100\\%,150\\%$ of the base premium for the Bonus-Malus levels $-2,-1,0,1$, respectively. That is, we let $p^{\\mathcal{B}\\mathcal{M}}(-2,t)=0.6p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(-1,t)=0.8p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(0,t)=p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, $p^{\\mathcal{B}\\mathcal{M}}(1,t)=1.5p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$ for all $t\\in\\{1,\\ldots,T\\}$. We fix the maximum compensation to be 1000, that is, $l^{\\mathcal{B}\\mathcal{M}}_{\\max}(b,t)=1000$ for all $b\\in\\mathcal{B},t\\in\\{1,\\ldots,T\\}$.\nWe set the deductible to be 0.5 for all but the final policy year, and set the deductible to be 5 for the final policy year, that is, $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,t)=0.5$ for all $b\\in\\mathcal{B},t\\in\\{1,\\ldots,T-1\\}$ and $l^{\\mathcal{B}\\mathcal{M}}_{\\mathrm{d}\\mathrm{t}\\mathrm{b}}(b,T)=5$ for all $b\\in\\mathcal{B}$. This is to prevent an issue caused by the finite horizon. Since after the final policy year there is no future benefit from the insurance policy and the insured is not incentivised to adopt the self-mitigation measure, a higher deductible is used as the incentive in the final policy year. \nIn addition, we let $\\delta_{\\mathrm{i}\\mathrm{n}}(t)=0.75(t-16)^+$, $\\delta_{\\mathrm{o}\\mathrm{u}\\mathrm{t}}(t)=3+\\frac{5}{19}(t-1)$, and $\\delta_{\\mathrm{r}\\mathrm{e}}=3$. This setting has the effect of incentivising the insured to activate the insurance contract early on, and disincentivising withdrawal when close to the final policy year. \nAs a baseline for comparison, we also consider another cyber risk insurance policy without the Bonus-Malus system, which can be modelled by letting $\\mathcal{B}=\\{0\\}$. We fix the premium to be the base premium $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}$, and leave everything else identical to the policy with the Bonus-Malus system. \n\n\\subsection{Results and Discussion}\n\nFigure~\\ref{fig:exp1policy} shows the expected number of years the insured's policy spends in each of the Bonus-Malus levels or being uninsured and the expected number of years the insured adopts the self-mitigation measure. The two panels compare the cyber risk insurance contract with the Bonus-Malus system with the one without.\nWith the contract that does not have the Bonus-Malus system, the decisions of the insured are completely deterministic, that is, they do not depend on the realisation of the cyber loss events. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.410$, the optimal strategy of the insured is to purchase the cyber risk insurance policy every year and only adopt the self-mitigation measure in the final policy year (due to the higher deductible in the final policy year). When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge4.415$, the optimal strategy of the insured is to never purchase the cyber risk insurance policy and always adopt the self-mitigation measure. Therefore, without the Bonus-Malus system, the issue of moral hazard is present and the insured will treat the cyber risk insurance policy and the self-mitigation measure as substitute goods. \nOn the other hand, when the Bonus-Malus system is introduced to the cyber risk insurance contract, the decisions of the insured depend on the realisation of the cyber loss events. When $4.495\\le p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.930$, the optimal strategy of the insured is to always purchase the cyber risk insurance policy and adopt the self-mitigation measure. When $4.935\\le p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le5.050$, the optimal strategy of the insured is to always adopt the self-mitigation measure but withdraw from the insurance contract when the expected future cost exceeds the expected future benefit of the insurance policy. As a result, the retention rate, i.e., the expected proportion of years the insured remains in the contract, drops when the base premium is increased. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge5.055$, the optimal strategy of the insured is to never purchase the cyber risk insurance policy and always adopt the self-mitigation measure. Hence, compared with the contract without Bonus-Malus, the contract with Bonus-Malus incentivises the insured to adopt the self-mitigation measure in addition to purchasing the cyber risk insurance policy. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_wo_BM1-eps-converted-to.pdf}\n~\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_w_BM1-eps-converted-to.pdf}\n\\caption{The retention rate of the cyber risk insurance policy and the expected years of adoption of the self-mitigation measure versus the base premium.}\n\\label{fig:exp1policy}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_wo_BM2-eps-converted-to.pdf}\n~\n\\includegraphics[width=0.48\\linewidth]{figures\/exp_ori_w_BM2-eps-converted-to.pdf}\n\\caption{The discounted total expected loss prevented by the self-mitigation measure and the discounted expected profit (measured by the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$) of the insurer versus the base premium. Left panel: the contract without the Bonus-Malus system. The dashed lines indicate the highest base premium before the insured chooses not to purchase the cyber risk insurance policy. Right panel: the contract with the Bonus-Malus system. The dashed lines indicate the highest base premium before the retention rate drops below 100\\%. The dotted lines indicate the highest base premium before the insured chooses not to purchase the cyber risk insurance policy.}\n\\label{fig:exp1profit}\n\\end{figure}\n\nFigure~\\ref{fig:exp1profit} compares both the expected value of the discounted total loss prevented by the self-mitigation measure and the expected value of the discounted profit of the insurer measured by the quantity $\\overline{Z}_{\\mathrm{i}\\mathrm{n}\\mathrm{s}}-\\overline{Z}_{\\mathrm{c}\\mathrm{p}}$ (defined in Section~\\ref{ssec:pricing}) in the two policies. The left panel of Figure~\\ref{fig:exp1profit} shows the case without Bonus-Malus. In that case, when $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\le4.410$, the insured will always purchase the cyber risk insurance policy but will only adopt the self-mitigation measure in the final policy year. Hence, the discounted total expected loss prevented stays at $0.505$, while the discounted expected profit of the insurer increases as the base premium increases. When $p^{\\mathcal{B}\\mathcal{M}}_{\\text{base}}\\ge4.415$, the insured will not purchase the insurance policy but will always adopt the self-mitigation measure. As a result, the discounted total expected loss prevented will be $17.183$ but the insurer will earn no profit. The most the insurer can gain before losing the insured is $-10.510$, when the base premium is set to $4.410$. \nIn contrast, in the case with the Bonus-Malus system, as shown in the right panel of Figure~\\ref{fig:exp1profit}, the insurer can gain a discounted expected profit of at most $-0.860$ while always retaining the insured (i.e., the insured will never withdraw from the contract), when the base premium is set to $4.930$. The insurer can gain a discounted expected profit of at most $-0.006$ before losing the insured, when the base premium is set to $5.050$. With both of these base premiums, the insured will always adopt the self-mitigation measure, resulting in a discounted total expected loss prevention of~$17.183$. \n\nOverall, this experiment demonstrates two benefits of the Bonus-Malus system. First, the presence of the Bonus-Malus system in the cyber risk insurance contract incentivises the insured to adopt the self-mitigation measure. This results in a considerable increase in the prevention of cyber losses, which enhances the overall security of the cyberspace. Second, the Bonus-Malus system benefits the insurer, since it allows the insurer to gain more profit from the cyber risk insurance policy while remaining attractive to the insured. \nTo show that the observations from the results in this experiment and the conclusions drawn do not depend on the specific choice of the $h$ parameter in the loss severity distribution (which is the most impactful parameter in the truncated g-and-h distribution), and that they also do not depend on our distributional assumption, we have repeated the same experiment with slightly modified settings.\nIn the first modified setting, the $h$ parameter in the truncated g-and-h distribution is set to $0.10$, $0.20$, or $0.25$. \nIn the second modified setting, the loss severity distribution is replaced by a log-normal distribution where the parameters are determined by matching the first two moments to $\\text{Tr-}g\\text{-and-}h(\\alpha=0,\\varsigma=1,g=1.8,h=0.15)$. \nThe results obtained under these modified settings turned out to be very similar to the results in the original experiment.\\footnote{The results under these modified settings are available in the online appendix on GitHub: \\url{https:\/\/github.com\/qikunxiang\/CyberInsuranceBonusMalus}}\nThis shows that the benefits of the Bonus-Malus system in the experiment are in fact present across various loss distributional assumptions.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThis paper motivated the joint consideration of risk reduction and risk transfer decisions in the face of cyber threats. We introduced a cyber risk insurance contract with a Bonus-Malus system to provide incentive mechanisms to promote the adoption of cyber risk mitigation practices. We developed a model based on the stochastic optimal control framework to analyse how a rational insured allocates funds between self-mitigation measures and a cyber risk insurance policy. A dynamic programming-based algorithm was then developed to efficiently solve this decision problem. \nA numerical experiment demonstrated that this novel type of insurance contract can incentivise the adoption of self-mitigation measures and can allow the insurer to profit more from the policy while remaining attractive to the insured. \nFuture research could investigate the effects of the risk profile, i.e., the characteristics of the loss distribution such as the heaviness of its tail, on the effectiveness of the Bonus-Malus system and how one can tailor Bonus-Malus-based insurance contracts for different risk profiles.  \n\n\n\\section*{Acknowledgments}\n\\noindent\nAriel Neufeld gratefully acknowledges the financial support by his Nanyang Assistant Professorship Grant (NAP Grant) \\emph{Machine Learning based Algorithms in Finance and Insurance}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and related Research}\n\nDiscrete-time queueing networks are used to model a variety of scenarios, ranging from traffic control over parallel computing to wireless communication. They are closely related to the canonical control system\n\\begin{equation}\n\t\\label{eq::usual_control}\n\tx_{t+1} = A x_t + B_t v_t + D_t w_t\n\\end{equation}\nwith some significant differences: i) The controls $v_t$ are binary in nature and linearly constrained by $C v_t \\leq c$, e.g. due to the interference properties of wireless channels. ii) The state lives on the discrete set $x_t \\in \\mathbb{N}^n$ where it exhibits no inertia ($A = I$). iii) And crucially, the matrices $B_t$ and $D_t$ behave stochastically, implying that the effect of a control decision is not certain. Together with the class of back-pressure control policies, those systems form a well investigated subclass of control problems.\n\nThe prototype back-pressure policy, that we will call the max-weight policy ($\\mathsf{MW}$), was first introduced in \\cite{Tassiulas1992}, where the authors also proved its much praised property of being \\textit{throughput optimal}. This means, that $\\mathsf{MW}$ can manage any load of traffic, provided this load can somehow be supported by the network topology.\nOver time, many variations of $\\mathsf{MW}$ where developed, e.g. to allow for a generalized control objective \\cite{Meyn2009} \\cite{Kasparick2018}, or to increase its performance in special cases like networks with input-queued switches or time-varying channels \\cite{Mckeown1999} \\cite{Neely2005}. Specific shortcomings of $\\mathsf{MW}$, like e.g. high end-to-end delay, where investigated in \\cite{Khan2009} \\cite{Subramanian2007} \\cite{Ying2011} and later partially remedied by \\cite{Neely2005} \\cite{Ying2011} \\cite{Huang2013} \\cite{Xiong2011}, using e.g. shortest path algorithms to reduce delay especially in low traffic scenarios.\n\nIn this paper, we propose a novel control policy that is predictive in nature and that we will call predictive network control ($\\mathsf{PNC}$). It can be regarded as a generalization of $\\mathsf{MW}$, since it contains $\\mathsf{MW}$ as a special case. But while $\\mathsf{MW}$ and all its derivations are \\textit{myopic}, i.e. only aim to improve the system state for the \\textit{immediate} next time slot, $\\mathsf{PNC}$ aims to improve the system state for \\textit{multiple} time slots up until a prediction horizon. This leads to the calculation of an entire optimal \\textit{trajectory} of control vectors. However, instead of applying the entire trajectory for the next few time slots, only the first control vector is applied to the system and the process repeats in the consecutive time slot. This allows the controller to react to any unforeseen changes in the control system \\cite{Mayne2000}.\n\nSuch a control scheme is called model-predictive control (MPC), and therefore $\\mathsf{PNC}$ is a realization of MPC, tailored specifically towards queueing networks. MPC itself is a well established branch of control theory and can cope very easily with hard constraints and non-linearities, making it particularly suited for our control problem. However, its advantages are payed for by high requirements on computational resources.\nSo far, there has only been one attempt to bring MPC to queueing networks: In \\cite{VanLeeuwaarden2010a} the authors focus on a special case of the standard model, in which only the arrivals to the system are of stochastic nature. The investigation is limited to numerical simulations, which show better system performance (smoother time behavior) for a designed MPC controller compared to simple feedback control laws.\nSince our queueing network will include a much higher degree of stochastics, we will not follow up on their work.\n\nBecause a queueing network misses any inertia ($A = I$), a predictive control scheme can only tap into its full potential, if the stochastics for $B_t$ (or $D_t$) are complex enough. E.g. if both matrices behave according to white-noise, prediction over more than the next time slot yields close to no improvement over myopic strategies. Hence, the benefit of a predictive control scheme usually increases with complexity of the system model. Therefore, we let $B_t$ (the matrix which is responsible for the topology and the quality of the links between the nodes of the network) be governed by a discrete-time Markov chain (DTMC) and a Bernoulli trial. This gives the opportunity to model long-term and short-term effects, respectively. Take e.g. wireless relay networks with user mobility: here, short-term interference leading to packet loss can be modeled by the Bernoulli trial, while long-term change in channel quality due to the mobility can be expressed by the DTMC \\cite{Guzman2019}.\n\nControl systems, in which the model parameters change according to a DTMC are called Jump-Markov systems (JMS). (Since simple feedback controllers cannot detect this change, JMS are usually controlled with MPC controllers). There exist several control approaches towards JMS, covering cases with linear \\cite{Park2002} \\cite{Chitraganti2014} \\cite{Tonne2017a} and even nonlinear system dynamics \\cite{Liu2015} \\cite{Tonne2017}, where the referenced works mainly differ in the choice of considered constraints.\nHowever, all these works deal with conventional control systems, where the controller usually tries to follow a reference trajectory and noise ($w_t$) represents a stochastic disturbance with zero first moment. In contrast, from the perspective of queueing networks, the noise term represents the arrival of packets\/customers whose first moment is strictly positive, and the controller tries to maintain finite queues for any given arrival (hence, there is no need for a reference trajectory). For that reason, prior work on JMS is only partially applicable to our systems. To the best of our knowledge, we are the first to consider both JMS and MPC in the context of discrete-time queueing networks.\n\nOur \\textbf{contribution} is three-folded: i) We develop a JMS-adapted discrete-time queueing network and introduce a family of predictive control policies, based on the paradigms of MPC. ii) We proof throughput optimality (the equivalent of stability) for the most simple of our predictive control policies, thereby implying the same for the rest. And iii) we show the benefit of these policies over the conventional back-pressure control ($\\mathsf{MW}$), using numerical simulation. In particular, our policies seem to maintain their throughput optimality in networks with synchronized queues, making them unique.\n\n\\section{System Model \\& Prerequisites}\n\n\\subsection{System Model}\n\nWe begin by stating the constituting equation for our system model and clarify its components afterwards. Similar to the conventional control system, a discrete-time queueing network can be expressed by its one-step evolution and associated constraints\n\\def\\hspace{0.5mm}{\\hspace{0.5mm}}\n\\begin{gather}\n    \\label{eq::basic_queueing}\n    q_{t+1} = q_t + R M_t v_t + a_t\n    \\\\[1ex]\n    \\nonumber\n    \\text{subject to}\n    \\\\[1ex]\n    \\nonumber\n    \\left(\n\t\\begin{aligned}\n\t    C v_t & \\leq c \\\\\n\t    - R^- v_t & \\leq q_t\n    \\end{aligned}\n    \\right)\n    \\hspace{0.5mm} \\text{and} \\hspace{0.5mm}\n    \\left(\n    \\begin{gathered}\n\t    M_t \\sim \\mathcal{B}(W^{s_t})\n\t    \\\\\n\t    W^{s_t} \\in \\{W^1 ,\\dots W^{n_s} \\}\n\t    \\\\\n\t    (s_t) \\sim \\operatorname{DTMC}(\\{1,\\dots n_s\\},P,s_0)\n    \\end{gathered}\n    \\right)\n\\end{gather}\nThe \\textit{queue vector} $q_t = \\left( q_t^1 \\dots q^{n_q}_t \\right)^\\intercal \\in \\mathcal{Q} = \\mathbb{N}^{n_q}$ represents the system- (or queue-) state, where $q_t^i$ counts the number of packets, waiting in queue $i = 1,\\dots n_q$ in time slot $t$. Each queue itself is a node of the network.\n\nIn any time slot, packets can be transmitted from one queue to another if there exists a directed link between the two and the link is activated.\nThere are $n_v$ links, each of which can be represented by a vector $r^j \\in \\left\\{ -1,0,+1 \\right\\}^{n_q}$ ($j = 1,\\dots n_v$), that, by superpositioning with $q_t$, transfers a packet from one queue ($\\{-1\\}$) to another ($\\{+1\\}$). All links are collected as columns in the routing matrix $R \\in \\left\\{ -1,0,+1 \\right\\}^{n_q \\times n_v}$ which therefore holds the topology.\n\n[Remark:\nIn \\textit{conventional} networks, a link has exactly \\textit{one} $\\{-1\\}$ entry (origin) and \\textit{at most one} $\\{+1\\}$ entry (destination). This implicit constraint is a prerequisite for all back-pressure policies to develop their throughput optimality. Though we will also use this constraint throughout the paper, our novel control policies seem to maintain their throughput optimality even when it is violated (see section \\ref{subsec::synchronized_queues}), allowing us to control networks with synchronized queues.]\n\nThe controller may activate a link in a given time slot via the binary control vector $v_t \\in \\{ 0 , 1 \\}^{n_v}$.\nIf we could activate all links simultaneously ($v_t = \\mathbf{1}_{n_v}$), the control problem would become trivial. However, we are usually constrained in the activation (e.g. due to interference properties) by the \\textit{constituency constraint} $Cv_t \\leq c$. The dimensions of $C$ and $c$ are case dependent, their entries are from the set $\\mathbb{N}$. Furthermore, a packet can only be scheduled for transmission, if it is present at the corresponding queue, hence a packet may only traverse a single link per time slot. We will refer to this as the \\textit{positiveness constraint}, which is readily implemented by considering the maximum one-step efflux of the system, which is $R^- v_t$, where $R^-$ is equal to $R$ without its positive entries. Naturally, the maximum efflux cannot drain more packets than are actually present: $q_t + R^- v_t \\geq 0$. Note that this also guarantees that $q_t \\in \\mathbb{N}^{n_q}$.\n\nFor clarification, we refer to \\figref{fig::min_example}. Here, we stated topology and constituency matrices and derived the corresponding constraints. \nGiven only $C$ and $c$, both components of $v_t$ could be active simultaneously. However, if $q^2$ is empty ($q^2 = 0$), it is not possible to activate the second link $r^2$.\n\n\\def\\comR{\n\t$\n\tR = \\begin{pmatrix}\n\t-1 & 0 \\\\ +1 & -1\n\t\\end{pmatrix}\n\t$\n}\n\\def\\comC{\n\t$\n\tC = \\begin{pmatrix}\n\t0 & 0\n\t\\end{pmatrix}\n\t$\n}\n\\def\\comc{\n\t$ c = 1 $\n}\n\\def\\comRC{\n\t$ \\begin{pmatrix}\n\t1 & 0 \\\\ 0 & 1 \n\t\\end{pmatrix} v_t \\leq q_t$\n}\n\\def\\comCC{\n\t$ \\begin{pmatrix}\n\t\t0 & 0 \n\t\\end{pmatrix} v_t \\leq 1$\n}\n\\def\\comA{\n\t$\\Downarrow$\n}\n\n\\begin{figure}[htbp]\n\t\\centering\n\n\t\\includegraphics[]{network_example_f.pdf}\n\t\\caption{Minimal Example of a Queueing Network}\n\t\\label{fig::min_example}\n\\end{figure}\n\n\nStill, even an activated link $r^j$ might fail in its transmission, leaving source and destination queue unchanged. This is modeled by a stochastic variable $m^j_t \\in \\{0,1\\}$ which is Bernoulli distributed (coin-flip) with probability $\\e{m}^j_t \\in [0,1]$. I.e. $m^j_t \\sim \\mathcal{B}(\\e{m}^j_t)$. For a succinct notation, we collect all those quantities in the diagonal matrices $M_t = \\operatorname{diag}_{j = 1 ,\\dots n_v}\\{m^j_t\\}$ and $\\e{M}_t = \\operatorname{diag}_{j = 1 ,\\dots n_v}\\{\\e{m}^j_t\\}$ respectively such that $M_t \\sim \\mathcal{B}(\\e{M}_t)$ and of course $\\E{M_t} = \\e{M}_t$. \n\nThe Bernoulli trials on $\\e{M}_t$ are intended to model short-term stochastics. For long-term stochastics, we let $\\e{M}_t$ be picked from a predetermined set $\\mathcal{W} = \\{ W^1,W^2,\\dots \\}$ of weight matrices $W^i$ according to a DTMC $(s_t)$. If $\\mathcal{S} = \\{1 ,\\dots n_s\\}$ is the index set of $\\mathcal{W}$, we have $(s_t) \\sim \\operatorname{DTMC}(\\mathcal{S},P,s_0)$ where $P$ and $s_0$ are transition matrix and initial state, respectively. The entire selection process can therefore be expressed as $\\e{M}_t = W^{s_t}$. If $\\sigma_t$ describes a distribution for the DTMC, we have $\\lim_{t\\to\\infty} \\sigma_t = \\pi$, which we assume to be the only stable distribution, with $\\pi^{\\R{s}}$ being the average probability of $s_t = {\\R{s}}$.\n\nThe task for a controller is to steer the packets through the network to their destination nodes. Once reached, the packets leave the system, which can be modeled via links $r^j$ without the $\\{+1\\}$ entry. At the same time, new packets are created directly at the queues through an \\textit{arrival vector} $a_t \\in \\mathbb{N}^{n_q}$ of possibly stochastic nature. We call $\\e{a} = \\E{a_t}$ the \\textit{arrival rate} and make the usual assumption, that there is an upper bound, such that always $a_t \\leq \\hat{a}$.\n\nFinally, we remark that if different packets are destined for different final destination nodes, they are of different \\textit{class} (or belong to a different \\textit{flow}). Each class has to have its separate network of queues in order for the packets to be distinguishable. Thus, for each class a new copy of the system would have to be employed. While many authors model this by adding an additional dimension (the dimension of all classes) to all quantities, we will just assume, that the so far described system model already consists of those copies, stacked in a suitable way, thereby avoiding the introduction of another dimension to the system model.\n\n\\subsection{Control Policies and Throughput Optimality}\n\nWe already defined $q_t \\in \\mathcal{Q}$ and $s_t \\in \\mathcal{S}$. Note that $(s_t)$ is a DTMC and $q_{t-1}$ is not needed for a prediction of future states once $q_t$ is known. If we further assume the arrival vector to be independent of past realizations, there is no reason for a controller to use any but the last known realizations of $q_t$ and $s_t$ for its decision making.\nIf we also define the set of all control vectors by\n\\begin{equation}\n\t\\mathcal{V} = \\Set{ v \\in \\{0,1\\}^{n_v} | \n\t\t\\begin{aligned}\n\t    C v & \\leq c \\\\\n\t   \n    \t\\end{aligned}\n    }\n\\end{equation}\nthen we can express a control policy $\\phi$ as a mapping from the set of relevant, observed quantities onto the set of control vectors: $\\phi: \\mathcal{Q} \\times \\mathcal{S} \\to \\mathcal{V}$. In some cases, however, it makes sense to incorporate a stochastic process into the policy itself, in order to circumvent the discreteness of $\\mathcal{V}$. This way, given a fixed pair of observations $(q',s')$, we can not only access a fixed $v' = \\phi(q',s') \\in \\mathcal{V}$, but on average \\textit{any} predetermined element $\\sum_{v \\in \\mathcal{V}} \\lambda^v v$ of the convex hull $\\conv{\\mathcal{V}}$. Hence we define a \\textbf{control policy} as\n\\begin{equation}\n\t\\phi: \\mathcal{Q} \\times \\mathcal{S} \\times \\Omega \\to \\mathcal{V}\n\\end{equation}\nwhere $\\Omega$ is the sample space of the underlying stochastic process. (Remember that a control policy is only valid, if it complies with $R^- \\phi(q_t,s_t,\\omega_t) \\leq q_t$.)\n\nWe say that a control policy $\\phi$ \\textbf{stabilizes} a system for a given arrival rate $\\e{a}$ if it can compensate the arrival rate on average:\n\\begin{equation}\n\\label{eq::def_stability}\n\\begin{aligned}\n\t\\mathbf{0} &= \\lim_{\\tau \\to \\infty} \\frac{1}{\\tau} \\sum_{t=1}^\\tau \\left( \\vphantom{\\frac{1}{2}} a_t + R M_t \\phi(q_t,s_t,\\omega_t) \\right)\n\t\\\\\n\t&= \\hspace{10mm} \\e{a} \\hspace{10.7mm} + \\sum_{\\R{s} \\in \\mathcal{S}} \\pi^\\R{s} RW^\\R{s} \\phi(q_t,\\R{s},\\omega_t)\n\\end{aligned}\n\\end{equation}\nComparing with the system equation \\eqref{eq::basic_queueing}, this implies that the average queue state remains bounded.\n\nFinally, a control policy $\\phi$ is throughput optimal, if it stabilizes a given system for any arrival rate $\\e{a}$ for which at least one other (possibly unknown) policy stabilizes the system. This can readily be expressed by noting that a policy $v_t = \\phi(s_t,\\omega_t)$ can on average, for every state of $\\mathcal{S}$ separately, excess any predetermined element in the interior of the convex hull $\\conv{\\mathcal{V}}$. Note that this excludes the boundary of $\\conv{\\mathcal{V}}$, because due to the positiveness constraints, no policy can guarantee to never be forced to be idle (meaning $v_t = \\mathbf{0}$). Naturally, there are no other options for the average control vector than those in $\\conv{\\mathcal{V}}$.\nThus, $\\phi$ is \\textbf{throughput optimal}, if it stabilizes the system for all $\\e{a}$ with\n\\begin{equation}\n\t\\label{eq::def_to}\n\t\\e{a} + \\sum_{\\R{s} \\in \\mathcal{S}} \\pi^\\R{s} RW^\\R{s} \\sum_{v \\in \\mathcal{V}} \\lambda^{\\R{s},v}  v = -\\varepsilon \\mathbf{1}_{n_q}\n\t\\hspace{10mm}\n\t\\begin{aligned}\n\t\\lambda^{\\R{s},v} & \\geq 0 \\\\\n\t\\sum_v \\lambda^{\\R{s},v} & \\leq 1\n\t\\end{aligned}\n\\end{equation}\nwhere $\\varepsilon > 0$ is used to exclude the boundary of $\\conv{\\mathcal{V}}$.\n\n\\section{Predictive Network Control ($\\mathsf{PNC}$)}\n\nInspired by the common MPC paradigms, our novel control policy, $\\mathsf{PNC}$, works in 3 steps: i) An entire trajectory of optimal control decisions from $t$ up until $t+H-1$ is calculated as the result of a minimization of an objective function $J$. Here, $J$ is a function of the next $H$ future system states, which we can only \\textit{predict}. $H$ is called the prediction horizon. ii) Only the first (i.e. the immediate) control decision in this trajectory is actually applied to the system. iii) The process repeats (discarding the rest of the just calculated trajectory). W.l.o.g., for the rest of the paper, we assume the current time slot to be $t=0$.\n\nThe most often encountered objective is the sum of squares, which in our case translates to\n\\begin{equation}\n\t\\label{eq::pnc_objective}\n\tJ(q_0,s_0) = \\CE{ \\sum_{t=1}^{H} q_t^Tq_t }{q_0,s_0}\n\\end{equation}\nUsing this definition, minimizing $J$ means minimizing the amount of packets in the network, which can only be done by delivering the packets to their destinations.\nOver the system evolution \\eqref{eq::basic_queueing}, $J$ will be influenced by the choice of control vectors via\n\\begin{equation}\n\t\\label{eq::actual_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s}\n    v_t\n    + T\\e{a}\n\\end{equation}\nwhere $\\sigma_0$ is the distribution corresponding to the initial state $s_0$ and $\\left( \\sigma_0 P^{(t)} \\right)^\\R{s}$ stands for the $\\R{s}$-th entry of the predicted distribution in time slot $t$. \n\nNote, that the prediction can be implemented in three different ways, varying in precision and required effort:\n\ni) The first one is the \\textit{true prediction}, which assigns a control vector to \\textit{every} time slot (up until $H$) and \\textit{every} possible set of realizations of the quantities in the system evolution. Since the ensemble of these realizations in time slot $t$ forms $q_{t+1}$ and $s_{t+1}$, this would mean making $v_t$ a function of $q_t$ and $s_t$ for the remainder of the prediction. The number of control vectors, required for such a prediction amounts to $H \\cdot n_s \\cdot k_q$, where $k_q$ is the number of all possible queue states, that can be realized in a single time slot (likely to depend itself on prior queue state, realization of arrival and Bernoulli trial). This obviously requires the maximum amount of computational resources but allows us to truly find the optimal control trajectory that minimizes $J$.\n\nii) In contrast, a \\textit{relaxed prediction} uses only a minimum of control vectors. I.e. in every time slot, a single control vector is chosen and thus $v_t$ is only a function of $t$ for the remainder of the prediction. (Using even less control vectors would not constitute a meaningful prediction for our purposes.) This amounts to only $H$ control vectors being required for the prediction, speeding up the calculation of an optimal control trajectory to minimize $J$ considerably. However, said trajectory might be sub-optimal compared to the true prediction from before and as a consequence the control performance might be worse.\n\niii) Finally, a mixture of both cases could be implemented, finding a balance between computational complexity and control performance. E.g. one could consider every future DTMC state $s_t$ (up until $s_H$) leading to $H \\cdot n_s$ control vectors that have to be determined in order to minimize $J$.\n\nWe will define or policy via case ii), i.e. the relaxed prediction and explain the reasoning for this in the end of the section. For what follows, we substitute the control vector $v_t$ with $u_t$ to emphasize, that this is not the actual control of the queueing network but rather the one used for the prediction. Hence, $u_t$ is a quantity that is used internally to define the $\\mathsf{PNC}$ policy according to\n\\begin{multline}\n\t\\label{eq::pnc_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s} u_t + T\\e{a}\n\\end{multline}\nIf we define the trajectory\n\\begin{equation}\n    \\label{eq::def_w_tilde}\n    \\tr{u}_0^{H-1} = \\begin{pmatrix}\n        u_0 \\\\\n        \\vdots \\\\\n        u_{H-1}\n    \\end{pmatrix}\n\\end{equation}\nand substitute it together with \\eqref{eq::pnc_expected_evolution} into \\eqref{eq::pnc_objective}, the objective can be rewritten as\n\\begin{equation*}\n    J(q_0,s_0) = J_1(q_0) + J_2(q_0,s_0) \\tr{u}_0^{H-1} + \\left( {\\tr{u}_0^{H-1}} \\right)^\\intercal J_3(s_0) \\tr{u}_0^{H-1}\n\\end{equation*}\nAs can be seen, $J_1(q_0)$ will not be influenced by the minimization over $\\tr{u}_0^{H-1}$, and $J_3(s_0)$ will stay bounded, since it is not dependent on $q_0$. Therefore, for large enough $q_0$, the linear term $J_2(q_0,s_0) \\tr{u}_0^{H-1}$ will always dominate the minimization over $\\tr{u}_0^{H-1}$, making it prudent to define our actual objective only over this linear term. This simplifies the minimization from a quadratic to a linear one. (Note that a similar step is also taken in the definition of the $\\mathsf{MW}$ policy.) Expanding the remaining constraints from the original system in a straight forward way, we end up with the following definition of the $\\mathsf{PNC}$ policy:\n\\begin{equation}\n\t\\label{eq::pnc_policy_def}\n\\begin{gathered}\n    \\phi^{\\mathsf{PNC}}(q_0,s_0) = \\operatorname{first} \\argmin_{\\tr{u}_0^{H-1}} \\  J_2(q_0,s_0) \\tr{u}_0^{H-1}\n    \\\\[2ex]\n    \\text{subject to} \\qquad\n    \\begin{aligned}\n    \\tr{C} \\tr{u}_0^{H-1} &\\leq \\tr{c}\n    \\\\\n    \\tr{A} \\tr{u}_0^{H-1} &\\leq \\tr{b}(q_0)\n    \\\\\n    \\tr{u}_0^{H-1} &\\in \\{0,1\\}^{Hn_v}\n    \\end{aligned}\n\\end{gathered}\n\\end{equation}\nwhere $\\operatorname{first} \\argmin ()$ expresses that only the first argument of the trajectory is used as output. An overview of the utilized quantities can be found in Table~\\ref{table_01}.\n\nChoosing $H=1$, we end up with the common $\\mathsf{MW}$ policy, which is not surprising, since its definition also involves a quadratic objective. And indeed, $\\mathsf{PNC}$ would merely be the extension of $\\mathsf{MW}$ over multiple time slots, if the $\\mathsf{PNC}$ controller would follow a once calculated optimal trajectory to its end (i.e. for $H$ time slots). However, $\\mathsf{PNC}$ recalculates this trajectory each time slot, thereby discarding its entire tail. This results in a much improved behavior of $\\mathsf{PNC}$ (see section \\ref{subsec::synchronized_queues}) but also makes it impossible to infer any properties from $\\mathsf{MW}$ to $\\mathsf{PNC}$. For more comparisons between the two policies, we refer to \\cite{Schoeffauer2018a} and \\cite{Schoeffauer2019}.\n\nWe continue with the main theorem of this paper, which states throughput optimality of the $\\mathsf{PNC}$ policy. Note that this automatically implies throughput optimality for every other MPC controller, that uses a more precise prediction (under the same constraints and objective function).\n\\begin{theorem}\n\\label{theorem::to_of_pnc}\nThe $\\mathsf{PNC}$ policy \\eqref{eq::pnc_policy_def} is throughput optimal for the system \\eqref{eq::basic_queueing}.\n\\end{theorem}\n\n\\begin{table*}[ht!]\n\\caption{Extended Formulas for the Optimization Problem}\n\\centering\n\\normalsize\n\\begin{tabular}{|p{0.3\\linewidth}|p{0.64\\linewidth}|}\n\\hline\n\\hline\n\\begin{center}\nExpected value of weight matrix $W^{s_t}$\n\\end{center}\n&\n\\begin{center}\nLinear objective $J_2$\n\\end{center}\n\\\\\n\\vspace{-2.8mm}\n\\begin{equation}\n\\begin{gathered}\n    \\e{W}_t(s_0) = \n\t\\CE{W^{s_t}}{s_0} \n\t\\\\  =\n    \\left( \\sigma_0 P^{(t)} \\otimes I_{n_s} \\right)\n    \\begin{pmatrix}\n\t\tW^1 \\\\ W^2 \\\\ \\vdots \\\\ W^{n_s}\n    \\end{pmatrix}^\\intercal\n    \\end{gathered}\n\\end{equation}\n&\n\\begin{equation}\n    \\label{eq::J2_formula}\n    J_2 = \n    2q_0^\\intercal R\n\t\\begin{pmatrix}\n\t\tH \\e{W}_0(s_0)\n\t\t\\\\\n\t\t(H-1) \\e{W}_1(s_0)\n\t\t\\\\\n\t\t\\vdots\n\t\t\\\\\n\t\t \\e{W}_{H-1}(s_0)\n\t\\end{pmatrix}^\\intercal\n\t+ \\e{a}^\\intercal R\n\t\\begin{pmatrix}\n\t\t(H+1)(H-0) \\e{W}_0(s_0)\n\t\t\\\\\n\t\t(H+2)(H-1) \\e{W}_1(s_0)\n\t\t\\\\\n\t\t\\vdots\n\t\t\\\\\n\t\t2H \\e{W}_{H-1}(s_0)\n\t\\end{pmatrix}^\\intercal\n\\end{equation}\n\\\\\n\\hline\n\\begin{center}\nConstituency constraints\n\\end{center}\n&\n\\begin{center}\nPositiveness constraints\n\\end{center}\n\\\\\n\\begin{equation}\n\t\\label{eq::table_constituency}\n\\underbrace{\n    \t\t\\left( \\vphantom{\\frac{1}{2}} I_{H} \\otimes C \\right) \n\t}_{\\displaystyle \\tr{C} }  \t\n \t\\tr{u}_0^{H-1} \\leq\n\t\\underbrace{ \t\n \t\t\\mathbf{1}_{H} \\otimes c \\vphantom{\\left( \\vphantom{\\frac{1}{2}} I_{H} \\otimes C \\right) }\n \t}_{\\displaystyle \\tr{c} \\vphantom{\\tr{C}} }\n\\end{equation}\n&\n\\vspace{-2.6mm}\n\\begin{equation}\n\t\\label{eq::table_positiveness}\n    \\underbrace{\n    \\begin{pmatrix}\n        R^- &  & &  \\\\\n        R & R^- &   &  \\\\\n        \\vdots &  & \\ddots \\\\\n        R & \\dots & R & R^-\n    \\end{pmatrix}\n    }_{\\displaystyle \\tr{A}}\n    \\tr{u}_0^{H-1}\n    \\leq \n    \\underbrace{\n    \\begin{pmatrix}\n    \tq_0 \\\\ q_0 + \\e{a} \\\\ \\vdots \\\\ q_0 + (H-1) \\e{a}\n    \\end{pmatrix}\n    }_{\\displaystyle \\tr{b}(q_0) }\n\\end{equation}\n\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{table_01}\n\\end{table*}\n\n\\section{Proof of Theorem \\ref{theorem::to_of_pnc}}\n\\label{sec::proof}\n\nWe will now prove, that $\\phi^\\mathsf{PNC}$ is throughput optimal. And in contrast to the usual stability-related proofs employed for MPC controllers, we will not rely on a terminal set.\n\n\\subsection{Preliminaries}\n\nIt will often become necessary to upper and lower-bound certain expressions. We will use $K_i \\in \\mathbb{R}_+$, $i\\in \\mathbb{N}$ to denote those bounds or variables, whose values are of no further interest and are obvious to calculate. Crucially, any $K_i$ will be independent of the initial system state $q_0$!\n\nWe will use gothic letters to express realizations of random variables, such that e.g. $\\R{s_t}$ is a realization of $s_t$, hence $\\R{s_t} \\in \\mathcal{S}$. And because the corresponding set of realizations will always be very clear from the context, we will use the succinct notation $\\sum_{\\R{s_t}}$ instead of $\\sum_{\\R{s_t} \\in \\mathcal{S}}$ for the sum of all realizations (as is needed for e.g. expressing the expectation).\n\nGiven a trajectory $\\tr{x}$ of control vectors of certain length, we use $\\tr{x} \\in \\mathcal{P}(q_0)$, to express that $\\tr{x}$ abides to the positiveness constraints $\\tr{A} \\tr{x} \\leq \\tr{b}(q_0)$, where $\\tr{A}$ and $\\tr{b}(q_0)$ are defined as in \\eqref{eq::table_positiveness}, expect for a possibly different value of $H$ (depending on the length of $\\tr{x}$). Analogue, $\\tr{x} \\in \\mathcal{C}$ will express, that $\\tr{x}$ abides to the constituency constraints as in \\eqref{eq::table_constituency}.\n\nFinally, we make the definitions $\\Delta_0^T := q_T - q_0$. Keep in mind, that analogue to $q_t$ being a function of all prior stochastics and controls, $\\Delta_0^T$ is of course a function of all stochastics and controls in the time slot $0,\\dots T-1$. With the definition $\\norm{q_t} := q_t^\\intercal q_t$ (which is not meant to be a norm) this gives raise to\n\\begin{equation}\n\t\\label{eq::to_sub_to_q0}\n\t\\norm{q_T} = \\norm{\\Delta_0^T} + \\norm{q_0} + 2q_0^\\intercal \\Delta_0^T\n\\end{equation}\nNote, that this decomposition corresponds to the one for the objective function $J$ and we can now restate the objective of the $\\mathsf{PNC}$ policy, $J_2$, as\n\\begin{equation}\n\t\\label{eq::to_objective_better}\n\tJ_2(q_0,s_0) = \\CE{ \\sum_{t=1}^{H} 2q_0^\\intercal \\Delta_0^t }{q_0,s_0}\n\\end{equation}\nWith this notation we formulate the next lemmas, needed for the proof.\n\n\\begin{lemma}\nThe difference $\\Delta_0^T$ between two queue states is bounded (element-wise) by\n\\begin{equation}\n\t\\label{eq::to_gen_bounds}\n\t- T n_v \\mathbf{1}_{n_q}\n\t\\leq\n\t\\Delta_0^T\n\t\\leq\n\tT n_v \\mathbf{1}_{n_q}  + T \\hat{a}\n\\end{equation}\nleading to\n\\begin{equation}\n\t\\label{eq::to_quad_to_lin}\n\t\\norm{q_0} + 2q_0^\\intercal  \\Delta_0^T\n\t\\leq\n\t\\norm{q_T} \n\t\\leq \\norm{q_0} + 2q_0^\\intercal  \\Delta_0^T + K_1\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBetween time slots $0$ and $T$ we have at best a constant efflux of $n_v$ or at worst a constant influx of $n_v + \\hat{a}$ packets per queue per time slot (since there are at most $n_v$ links to fill or drain any given queue).\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma::diff}\nThe difference between the minimization that originates from the definition of the $\\mathsf{PNC}$ policy (using the formulation from \\eqref{eq::to_objective_better}), and the same minimization but without considering any positiveness constraints can be bounded by\n\\begin{gather}\n\\label{eq::to_lemma}\n\\begin{aligned}\n    \\min_{ \\tr{u}_0^{H-1} \\in \\mathcal{C} \\cap \\mathcal{P}(q_0) }\n    \t&\\CE{ \n    \t\t\\sum_{t=1}^{H}  2q_0^\\intercal \\Delta_0^t\n    \t}\n    \t{\n    \t\tq_0,s_0\n    \t}\n    \\\\\n    -\n    \\min_{ \\tr{u}_0^{H} \\in \\mathcal{C} } \\hspace{6mm}\n    \t&\\CE{ \n    \t\t\\sum_{t=1}^{H}  2q_0^\\intercal \\Delta_0^t\n    \t}\n    \t{\n    \t\tq_0,s_0\n    \t}    \n\\end{aligned}\n    \\\\ \\nonumber\n    \\leq\n    \\left( Hn_v^2-n_v \\right) \\left( H + 1 \\right)\n    = K_2\n\\end{gather}\n\\end{lemma}\n\\begin{proof}\nClearly, the maximum deficit, that $\\Delta_0^t$ can generate is (element-wise) less than $t n_v \\mathbf{1}_{n_q}$ (all links drain a queue over $t$ steps).\nFor a single queue, the \\textit{most} efflux in $H$ time slots is $Hn_v$ packets. \nHence, if $q_0 \\geq Hn_v \\mathbf{1}_{n_q}$, a control trajectory \\textit{cannot} violate the positiveness. Conversely, if a link cannot be activated due to the positiveness constraints, at least one entry of $q_0$ must be smaller than $H n_v$.\n\nDue to the linearity, the largest difference in the minimizations will be found, if $q_0 = (H n_v - 1) \\mathbf{1}_{n_q}$ (possibly denying any activation for the minimization with the positiveness constraints). This, together with the initial bound on $\\Delta_0^t$ leads directly to\n\\begin{equation}\n\t\\sum_{t=1}^H 2 (H n_v - 1) \\mathbf{1}_{n_q}^\\intercal  t n_v \\mathbf{1}_{n_q} = \\left( Hn_v^2-n_v \\right) \\left( H + 1 \\right)\n\\end{equation}\nwhich is an upper bound for the difference in question.\n\\end{proof}\n\nFinally, the following theorem will allow us to express our definition of stability by the means of a Ljapunov function.\n\\begin{lemma}\n\t\\label{lemma::foster}\n\tA policy $\\phi$ stabilizes the system \\eqref{eq::basic_queueing} under a certain arrival rate $\\e{a}$, if we can find a function $f: \\mathcal{Q} \\times \\mathcal{S} \\to \\mathbb{R}_+$ with the property\n\t\\begin{equation}\n\t\t\\label{eq::to_drift}\n    \t\\CE{ f(q_{1},s_{1}) - f(q_0,s_0) }{q_0} \\leq K_4 - K_5 \\mathbf{1}_{n_q}^\\intercal q_0\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWithout further ado, we take expectations and sum \\eqref{eq::to_drift} over multiple time slots to obtain the following sequence of arguments\n\\begin{gather}\n\t\\nonumber\n    \\E{f(q_T,s_T)} - \\E{f(q_0,s_0)}\n    \\leq T K_4 - K_5 \\sum_{t = 0}^{T - 1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t }\n\\\\\n\t\\nonumber\n    \\Longrightarrow \\qquad\n    -\\E{f(q_0,s_0)} \n    \\leq T K_4 - K_5 \\sum_{t = 0}^{T - 1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t }\n\t\\hphantom{\\Longrightarrow \\qquad}    \n\\\\\n    \\Longrightarrow \\qquad\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\E{ \\mathbf{1}_{n_q}^\\intercal q_t } \n    \\leq\n    \\frac{K_4}{K_5} + \\frac{\\E{f(q_0)}}{T K_5}\n    \\hphantom{\\Longrightarrow }\n\\\\\n\t\\nonumber\n    \\Longrightarrow \\qquad\n    \\lim_{T\\to\\infty}\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\E{ q_t } \\leq\n    \\frac{K_4}{K_5} \\mathbf{1}_{n_q} \n    \\hphantom{\\Longrightarrow \\qquad}\n\\end{gather}\nSince $\\E{q_t} \\geq \\mathbf{0}$ always, and the difference between consecutive states is bounded, all elements of the sequence $(\\E{q_t})$ must be bounded. From there, the stability condition \\eqref{eq::def_stability} follows immediately. (Note that $\\E{\\cdot}$ is the expectation of the stochastic in the system model and not over time.)\n\\end{proof}\n\n\\subsection{Main Proof}\n\nWe now start with the main part of the proof. We will define a Ljapunov function $f(q_t,s_t)$ and show that \\textit{if} the system is governed by the $\\mathsf{PNC}$ policy, $f$ fulfills Lemma \\ref{lemma::foster} for any possibly stabilizable arrival rate (see \\eqref{eq::def_to}).\n\nFor a $\\mathsf{PNC}$ policy with horizon $H+1$, we employ the following Ljapunov function:\n\\begin{equation}\n    \\label{eq::to_f_tilde}\n\\begin{gathered}    \n    f(q_0,s_0) =\n    \\min_{ \\tr{z}_0^{H-1} \\in \\mathcal{C}_z } \n    \\CE{ \n    \t\t\\sum_{t=1}^{H} \\norm{q_{t}}\n    }{q_0,s_0}\n\\end{gathered}\n\\end{equation}\nA few remarks are in order: i) The minimization in $f$ mimics the $\\mathsf{PNC}$ policy, but is in fact independent of it. ii) The horizon of the minimization of $f$ is chosen to be one step smaller than that of the $\\mathsf{PNC}$ policy. iii) The control vectors are now denoted by $z$ instead of $v$ or $u$, because they run independent of the actual control $v_t$ or the predicted control inside the $\\mathsf{PNC}$ controller $u_t$. Crucially, the control trajectory $\\tr{z}_0^{H-1}$ is state sensitive regarding the DTMC $(s_t)$ of the weight matrices and is \\textit{not} constraint by the positiveness constraints $\\mathcal{P}$. I.e.\n\\begin{multline}\n\t\\label{eq::to_expected_evolution}\n    \\CE{q_{T}}{q_0,\\sigma_0}\n    = q_0 + \\sum_{t = 0}^{T-1} \\sum_{\\R{s} \\in \\mathcal{S}} \\left( \\sigma_0 P^{(t)} \\right)^\\R{s} R W^\\R{s} z_t(\\R{s}) + T\\e{a}\n\\end{multline}\n\nThis last point is important: the minimization of the actual $\\mathsf{PNC}$ policy (with horizon $H+1$) uses the control trajectory $\\tr{u}_0^{H}$, which assigns to each time slot of the prediction \\textit{a single} control vector $u_t \\in \\mathcal{V}$. In contrast and per definition, the minimization in the Ljapunov function $f$ uses the control trajectory $\\tr{z}_0^{H-1}$, which assigns to each time slot \\textit{and} each possible realization of $s_t$ a control vector $z_t(s_t) \\in \\mathcal{V}$. For succinct notation we define $z_t^\\R{s} := z_t(s_t = \\R{s})$. The trajectory of the control vectors $z_t^\\R{s}$ is defined by first stacking over all realization of $(s_t)$ and then over all time slots:\n\\begin{equation}\n\t\\tr{z}_0^{H} \n\t=\n\t\\begin{pmatrix}\n\t\tz'_0 \\\\\n\t\tz'_1 \\\\\n\t\t\\vdots \\\\\n\t\tz'_H\n\t\\end{pmatrix}\n\t,\\qquad \\text{with} \\qquad\n\tz'_t =\n\t\\begin{pmatrix}\n\t\tz_t^1 \\\\\n\t\tz_t^2 \\\\\n\t\t\\vdots \\\\\n\t\tz_t^{n_s}\n\t\\end{pmatrix}\n\\end{equation}\n\nThe constraint $\\tr{z}_0^{H-1} \\in \\mathcal{C}_z$ in the definition of $f$ expresses, that each single control vector $z_t^\\R{s}$ is constraint by the constituency in the usual way ($C z_t^\\R{s} \\leq c$), and therefore $\\mathcal{C}_z$ is a straight forward expansion of $\\mathcal{C}$.\n\nWe can now start expressing the first term of \\eqref{eq::to_drift} (for now conditioning on $s_0$ as well) as\n\\begin{gather}\n\t\\label{eq::to_simple_first_term}\n\\CE{ f (q_1,s_1) }{ q_0,s_0 }\t =\n\t\\CE{\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C} }\n\t\t\\CE{\n\t\t\t\\sum_{t=2}^{H+1} \\norm{q_t}\n\t\t}{\n\t\t\tq_1 , s_1\n\t\t}\n\t}{\n\t\tq_0,s_0\n\t}\n\\end{gather}\nNote that this term is shifted in time. The control $v_0$, that leads from $q_0$ to $q_1$ is exactly the control, that is defined by the control policy $v_0 = \\phi^\\mathsf{PNC}$ and that actually affects the network. In contrast, the dummy controls $\\tr{z}_1^H$ are part of the function $f$, do not affect the actual network and therefore are independent of the chosen policy.\n\nUsing \\eqref{eq::to_sub_to_q0} the inner term of \\eqref{eq::to_simple_first_term} can become\n\\begin{align*}\n\t\\sum_{t=2}^{H+1} \\norm{q_t}\n\t&=\n\t\\sum_{t=2}^{H+1} \\Big(\n\t\t\\norm{q_0} + \\norm{\\Delta_0^t} + 2q_0^\\intercal \\Delta_0^t\n\t\\Big)\n\t\\\\\n\t&\\leq\n\tH \\norm{q_0} + K_6\n\t+\n\t\\sum_{t=2}^{H+1} \\Big(\n\t\t2q_0^\\intercal \\Delta_0^t\n\t\\Big)\n\\end{align*}\nThe individual sums of $\\Delta_0^t$ can be bounded according to \\eqref{eq::to_gen_bounds} by some constant $K_6$ which is unaffected by the minimization or the expectation from \\eqref{eq::to_simple_first_term}. The same holds for $\\norm{q_0}$ if we notice, that conditioning on $q_1$ is the same as conditioning on $q_0$ \\textit{and} $\\Delta_0^1$, since $q_1 = q_0 + \\Delta_0^1$. Hence, both terms can be pulled to the left-hand-side of \\eqref{eq::to_simple_first_term}, as seen in the first two lines of \\eqref{eq::to_long}. In what follows, we will step by step dissolve the outer expectation and expand the sum, which is possible, since the minimization is linear in $\\tr{z}_1^H$ and the constraints act on each control vector separately:\n\\begin{align}\n\t\\label{eq::to_long}\n\t& \\hspace{5mm}\n\t\\CE{ f (q_1,s_1) }{ q_0,s_0 } - H \\norm{q_0} - K_6 \n\\\\ \n\t\\nonumber\n\t& \\leq \n\t\\mathbb{E} \\Bigg[\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_1 , s_1\n\t\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\\mathbb{E} \\Bigg[\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z}\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0 ,\\Delta_0^1, s_1\n\t\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\\sum_{\\R{q_0},\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\t\\R{q_0} , \\R{s_1}\n\t\t\\Bigg]\n\t\\CP{\n\t\t\\R{q_0} , \\R{s_1}\n\t}{\n\t\tq_0,s_0\n\t}\n\\\\\n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{q_0},\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR M_t z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\t\\R{q_0} , \\R{s_1}\n\t\t\\Bigg] &\n\t\\\\\n\t\\cdot\t\n\t\\CP{\n\t\t\\R{q_0}\n\t}{\n\t\tq_0\n\t}\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} \\, &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR M_t z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0 , \\R{s_1}\n\t\t\\Bigg] &\n\t\\\\\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} \\, &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\\sum_{\\R{s_1}}\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2 q_0^\\intercal  \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{\\R{s_1}} &\n\t\\\\[-2ex]\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& \\hspace{1mm}\n\t\\begin{aligned}\n\t=\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{\\R{s_1}}\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2q_0^\\intercal  \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{\\R{s_1}} &\n\t\\\\[-2ex]\n\t\\cdot\n\t\\CP{\n\t\t\\R{s_1}\n\t}{\n\t\ts_0\n\t} &\n\t\\end{aligned}\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\\sum_{\\R{s_t}}\t\t\t\n\t\t\t\t2q_0^\\intercal   \\left(\n\t\t\t\tR W^\\R{s_t} z_t^\\R{s_t} + a_t \\right)\n\t\t\\CP{ \\R{s_t} }{s_0}\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\tau = t}^{H}\t\n\t\t\t\t2q_0^\\intercal   \\left(\n\t\t\t\tR M_{t} z_t^{s_t} + a_t \\right)\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& = \n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=2}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_1^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n\t\\nonumber\n\t& =\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_0^t\n\t\t\t\t- 2q_0^\\intercal  \\Delta_0^1\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\\\\ \n    \\label{eq::to_last_long}\n\t& \\leq\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z}\n\t\t\\mathbb{E} \\Bigg[\n\t\t\t\\sum_{t=1}^{H+1}\n\t\t\t\t2q_0^\\intercal  \n\t\t\t\t\\Delta_0^t\n\t\t\\, \\Bigg| \\,\n\t\t\tq_0,s_0\n\t\t\\Bigg]\n\t-\n\t\\min_{ \\tr{z}_0^{0} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\t\n\t2q_0^\\intercal  \\Delta_0^1\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{align}\nThough $\\Delta_0^1$ is steered by $v_0$ (the actual control of the system), every policy has to abide to the constituency, which allows for the last term to be formulated over $\\tr{z}_0^0 \\Leftrightarrow z_0^{s_0}$. Still, the first term of \\eqref{eq::to_last_long} depends on $v_0$ through $\\Delta_0^t$ so that we can rewrite it as\n\\begin{multline}\n\\label{eq::to_elim_a}\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H+1}\n\t\t\t2q_0^\\intercal \n\t\t\t\\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t=\n\t\\frac{ \\left( H+1 \\right) \\left( H+2 \\right) }{2} \\e{a}\n\\\\\n\t\t+\n\t\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal R \\left[\n\t\t    \\sum_0^H\n\t\t        W^{s_0} v_0\n\t\t    +\n\t\t    \\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\right]\n\\end{multline}\n\nTo interface the $\\mathsf{PNC}$ policy \\eqref{eq::pnc_policy_def}, we need to incorporate the positiveness constraints. To that end, we introduce a transformation in variables, centered around the idea, that each set $\\{ z_t^1,\\dots z_t^{n_s} \\}$ can be expressed by a common part $\\mu_t$, and $n_s$ differences $\\delta_t^i$:\n\\begin{equation}\n\\begin{gathered}\n\tz_t^i = \\mu_t + \\delta_t^i \\in \\mathcal{C}_z\n\t\\\\\n\t\\text{for} \\qquad  t = 1,\\dots H \\qquad \\text{and} \\qquad i = 1 ,\\dots n_s\n\t\\\\\n\t\\text{with}  \\qquad  \\mu_t,\\delta_t^i \\in \\{0,1\\}^{n_v} \\qquad \\text{and} \\qquad C \\mu_t \\leq c \n\\end{gathered}\n\\end{equation}\nWe define a suitable stacking of these new variables in such a way that\n$\\tr{z}_1^H = \\tr{\\mu}_1^H + \\tr{\\delta}_1^H$ and write $\\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H $, to express that $\\tr{\\delta}_1^H$ has to abide to usual constituency, if $\\tr{\\mu}_1^H$ has already been chosen (i.e. for each $\\delta_t^i$ separately it must hold that $C \\delta_t^i \\leq c - C \\mu_t$).\n\nNext, we substitute these variables into the last term of \\eqref{eq::to_elim_a}.\n\\def\\hspace{1cm}{\\hspace{1cm}}\n\\def\\hspace{3mm}{\\hspace{3mm}}\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n    \\nonumber\n    &\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal  R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H}  W^{s_0} v_0\n\t\t +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\mu}_1^H + \\tr{\\delta}_1^H \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal  R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\left( \\mu_t + \\delta_t^\\R{s_t} \\right) \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\left( \\sum_{\\R{s_t}} W^\\R{s_t} \\CP{\\R{s_t}}{s_0} \\right) \\mu_t \n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\delta_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\t\t\\\\ \\nonumber\n\t\t& \\hspace{1cm} +\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\e{W}_t(s_0) \\mu_t\n\t\t\\\\ \\nonumber\t\n\t\t& \\hspace{1cm} +\t\t\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} \\delta_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\leq\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H}  \\left( \\mathbf{0}_{n_v}^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} \\cap \\mathcal{P}(q_0)}\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\overset{\\mathclap{\\mathsf{PNC}}}{=}\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\left( v_0^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} \\cap \\mathcal{P}(q_0) }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\overset{\\mathclap{\\text{Lemma \\ref{lemma::diff}}}}{\n\t\\leq\n\t}\n\tK_7 +\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\left( v_0^\\intercal , \\left( \\tr{\\mu}_1^H\\right )^\\intercal \\right)^\\intercal \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t\\leq\n\tK_7 +\n\t\\min_{ \\tr{\\delta}_1^H \\in \\mathcal{C}_z \\setminus \\tr{\\mu}_1^H }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H}  v_0 \\in \\mathcal{C} }\n\t\\hspace{3mm}\n\t\\min_{ \\vphantom{\\tr{\\delta}_1^H} \\tr{\\mu}_1^H \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\cdot\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\tK_7 +\n\t\\min_{ \\tr{z}_1^H \\in \\mathcal{C}_z }\n\t\\hspace{3mm}\n\t\\min_{ v_0 \\in \\mathcal{C} }\n\t\t2q_0^\\intercal R \n\t\t\\Bigg[\n\t\t\\sum_{0}^{H} W^{s_0} v_0\n\\\\ \\nonumber\n\t\t& \\quad +\t\t\n\t\t\\sum_{t=1}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\t\t\\Bigg]\n\\\\ \\nonumber\n\t&\n\t=\n\tK_7 + \n\t\\min_{ \\tr{z}_0^H \\in \\mathcal{C}_z }\n\t2q_0^\\intercal R \n\t\t\\sum_{t=0}^H \\sum_{\\tau=t}^{H} \\sum_{\\R{s_t}} W^\\R{s_t} z_t^\\R{s_t} \\CP{\\R{s_t}}{s_0}\n\\\\\n    \\label{eq::to_intro_PNC}\n\t&\n\t=\n\tK_7 + \n\t\\min_{ \\tr{z}_0^H \\in \\mathcal{C}_z }\n\t\t\\CE{ \\sum_{t=1}^{H+1} 2q_0^\\intercal R \\Delta_0^t }{q_0,s_0}\n\\end{align}\nNote that we used the fact that $\\mu_t$, once separated from the $\\delta_t^i$ in a suitable manner, can be identified with the dummy control $u_t$ from the definition of the $\\mathsf{PNC}$ policy \\eqref{eq::pnc_objective}. Hence, the equality marked with the $\\mathsf{PNC}$ label \\textit{only} holds under the $\\mathsf{PNC}$ policy, since it chooses $v_0$ in such a way that the entire first term is minimized over the trajectory $\\tr{z}_0^H$ instead of only $\\tr{z}_1^H$.\n\nIf we now combine the results of \\eqref{eq::to_long}, \\eqref{eq::to_elim_a} and \\eqref{eq::to_intro_PNC} we get\n\\begin{gather}\n\t\\label{eq::to_first_drift_term}\n\t\\CE{ f (q_1,s_1) }{ q_0,s_0 } - H \\norm{q_0} - K_8 \n\t\\\\\n\t\\nonumber\n\t\\leq\n\t\\min_{ \\tr{z}_0^{H} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H+1}   2q_0^\\intercal  \\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t-\n\t\\min_{ \\tr{z}_0^{0} \\in \\hphantom{\\mathcal{C}} \\mathclap{\\mathcal{C}_z} }\n\t\\mathbb{E} \\Bigg[\t\n\t2q_0^\\intercal  \\Delta_0^1\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gather}\nwhich completes the derivation for the first term of \\eqref{eq::to_drift}.\n\nIn a similar but much easier fashion, the second term of \\eqref{eq::to_drift} can be reshaped into\n\\begin{equation}\n\t\\label{eq::to_second_drift_term}\n\\begin{gathered}\n\t\\CE{ f (q_0,s_0) }{ q_0,s_0 } - H \\norm{q_0}\n\t\\\\\n\t\\geq\n\t\\min_{ \\tr{z}_0^{H-1} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t\\sum_{t=1}^{H}   2q_0^\\intercal  \\Delta_0^t\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gathered}\n\\end{equation}\nCombining \\eqref{eq::to_first_drift_term} and \\eqref{eq::to_second_drift_term} results in\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0,s_0 }\n\\\\\n\t\\leq\n\tK_{8} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\\end{gather*}\nTo alleviate the outer conditioning on $s_0$ we take the expectation $\\CE{\\cdot}{q_0}$ on both sides, conditioned only on $q_0$, and swap minimization and expectation operator:\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\\\\\n\t\\leq\n\tK_{9} +\n\t\\mathbb{E} \\Bigg[\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0,s_0\n\t\\Bigg]\n\t\\, \\Bigg| \\,\n\t\tq_0\n\t\\Bigg]\n\\\\\n\t\\leq\n\tK_{9} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\\mathbb{E} \\Bigg[\n\t\t2q_0^\\intercal  \\Delta_1^{H+1}\n\t\\, \\Bigg| \\,\n\t\tq_0\n\t\\Bigg]\n\\end{gather*}\nFinally, recall that for throughput optimality, this expression has to be negative for each $\\e{a}$ that can be expressed via \\eqref{eq::def_to}. Substituting this we obtain\n\\begin{gather*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\\\\\n\\begin{aligned}\n\t& \\leq\n\tK_{9} +\n\t\\min_{ \\tr{z}_1^{H} \\in \\mathcal{C}_z }\n\t\t2q_0^\\intercal \\left(\n\t\t\\sum_{t=1}^{H}\n\t\t\t\\sum_{\\R{s}}  \\pi^\\R{s} R  W^\\R{s} z_t^\\R{s} + H \\e{a}\n\t\t\\right)\n\\\\ &\n\\begin{multlined}\n\t\\leq\n\tK_{9} +\n\t\\min_{ z' \\in \\mathcal{C}_z }\n\t\t2Hq_0^\\intercal \\Bigg(\n\t\t\\sum_{\\R{s}} \\pi^\\R{s} RW^\\R{s} z^\\R{s}\n\t\\\\\n\t\t-  \\varepsilon \\mathbf{1}_{n_q}\n\t\t-  \\sum_{\\R{s}} \\pi^\\R{s} R W^\\R{s} \\sum_{v \\in \\mathcal{V}} \\lambda^{\\R{s},v} v\n\t\t\\Bigg)\n\\end{multlined}\t\n\\end{aligned}\t\n\\end{gather*}\nBecause the minimization is linear, the optimum is found on the boundary and thus the first term in the bracket (which is subject to minimization) will at least cancel out the last term, leaving us with\n\\begin{align*}\n\t\\CE{ f (q_1,s_1) - f (q_0,s_0) }{ q_0 }\n\t& \\leq\n\tK_{9}\n\t-\n\t2 H \\varepsilon q_0^\\intercal  \\mathbf{1}_{n_q}\n\t\\\\\n\t& \\leq\n\tK_{9}\n\t-\n\tK_{10} \\mathbf{1}_{n_q}^\\intercal q_0 \n\\end{align*}\nwhich fulfills lemma \\ref{lemma::foster} and therefore proves throughput optimality of our $\\mathsf{PNC}$ policy.\n\\\\\n\\rightline{$\\Box$}\n\n\\section{Exemplary Applications of PNC}\n\\label{sec::examples}\n\n\\subsection{Dynamic Topology}\n\nWe employ a scenario as depicted in \\figref{fig::ext_1_scenario}, where a mobile user equipment (UE) crosses multiple sectors, each one designated to a specific access point (AP). In each sector, the UE can only communicate with the corresponding AP. The APs are connected to a global network from which they receive packets that they are supposed to transmit to the UE.\nThe derived queueing network is shown in \\figref{fig::ext_1_model}. We use a most simplified model to yield easily interpretable results: First, the DTMC is deterministic which allows us to fix the time behavior of the transmission success probabilities $\\e{m}^j_t$ of the links. Second, we model this deterministic time behavior as binary sequences which are depicted in \\figref{fig::ext_1_links}. This corresponds to the case, in which the UE travels with constant velocity along a known path and the sectors do not overlap. The UE remains in each sector for exactly 3 time slots, where it experiences perfect channel quality (guaranteed transmission success).\nFurther we assume that a single packet is created every second time slot at $q^1$, which represents the entire arrival to the system.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_f.pdf}\n  \\caption{Extension 1 - Scenario}\n  \\label{fig::ext_1_scenario}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_model_f.pdf}\n  \\caption{Example 1 - Queueing Network}\n  \\label{fig::ext_1_model}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_links_f.pdf}\n  \\caption{Example 1 - Link Probabilities}\n  \\label{fig::ext_1_links}\n\\end{figure}\n\nThe simulation results in \\figref{fig::ext_1_simu_2} depict the accumulated amount of packets send (blue) and received by the UE (green and red). For visualization purposes, we averaged the resulting step functions, so that they are presented as lines. It can be observed that only around 33\\% of the packets reach the UE for the conventional back pressure policy, $\\mathsf{MW}$, (red). (Note that $\\mathsf{MW}$ can be expressed as a special case of the $\\mathsf{PNC}$ policy, in which the horizon is $H=1$.) The other 66\\% remain at already past base stations. This high packet loss is due to $\\mathsf{MW}$ requiring time to establish its throughput optimality. Indirectly, $\\mathsf{MW}$ functions by using misplaced packets as an indicator for later control decisions. The presented example, however, is based on a transient event where this indicator function of misplaced packets is only of limited use.\n\nAs can be seen, using the novel $\\mathsf{PNC}$ with horizon $H=2$, (the lowest green line) already nearly doubles the amount of packets that arrive at the UE to 60\\%. For $H=5$ we reach 80\\%, a significant performance boost.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_DTMC_simu_f.pdf}\n  \\caption{Example 1 - Simulation with Multiple Policies}\n  \\label{fig::ext_1_simu_2}\n\\end{figure}\n\n\\subsection{Networks with Synchronized Queues}\n\\label{subsec::synchronized_queues}\n\nThe following application is motivated by fact, that though $\\mathsf{MW}$ performs poorly in networks with dynamic topology, it is still able to achieve throughput optimality in the long run, if we assume $\\mathsf{MW}$ to be sensitive to the current state of the DTMC. And it is only fair to make this assumption, since we assume the same for our $\\mathsf{PNC}$ policy. Hence, throughput optimality seems to be shared by both policies, if we talk about conventional networks. However, in the next example, we forgo conventional networks and introduce \\textit{synchronized} queues. In networks with synchronized queues, only $\\mathsf{PNC}$ seems to maintain its throughput optimality while $\\mathsf{MW}$ fails, giving a strong incentive to employ the $\\mathsf{PNC}$ policy.\n\nQueues are \\textit{synchronized} (or \\textit{paired}), if they can only be served at the same time. This can be useful, if one wants to\nexploit constructive interference \\cite{Timotheou2016} or\nmodel parallel processing tasks in computing \\cite{Evdokimova2018} and social matchmaking \\cite{Buke2015}.\nWhile synchronized queues have been studied on their own \\cite{Harrison1973} \\cite{Fayolle1979} \\cite{Borst2008} \\cite{DeCuypere2014}, there has not been any research on how they behave in a network. Indeed, \\cite{Schoeffauer2018a} presents a simple example, proving that $\\mathsf{MW}$ loses its throughput optimality if the network contains only a single pair of synchronized queues. The reason for that can be found equation A.18 from the original proof in \\cite{Tassiulas1992}, which loses its generality. In layman's terms, the original proof is based on the fact, that the evolution of the queue vector constitutes a DTMC by itself. And for conventional networks, there exists a \\textit{finite} set of states (of that DTMC) which can be shown to be recurrent. This makes the entire DTMC recurrent which corresponds to throughput optimality. However, introducing synchronized queues, the finite set grows to infinite size, invalidating this correspondence. This leads to the questions, in how far back-pressure policies are suited for such networks and if there exists another policy, which guarantees throughput optimality.\n\nTo illustrate that $\\mathsf{PNC}$ might be that policy, we refer to the example, depicted in \\figref{fig::constructive_interference_set_up}. Set-up and thereof derived queueing network are shown on the left and right side, respectively. The example consists of an access point (AP) $q_1$ that can either transmit solitary (link $r^1$), or initiate synchronized transmission (link $r^3$) with a neighboring AP $q_2$. The synchronized transmission uses constructive interference and thus achieves higher throughput. However, before synchronized transmission can be initiated, the data packets have to be shared (link $r^2$), i.e. copied from $q_1$ to $q_2$.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{motivation_LT_f.pdf}\n  \\caption{Example 2 - Scenario and Queueing Network}\n  \\label{fig::constructive_interference_set_up}\n\\end{figure}\n\nFor comprehensiveness, we use a constant success probability matrix $\\e{M}$, i.e. we do not make use of an DTMC to select different matrices from $\\mathcal{W}$. The $\\e{m}^j$ (which are the diagonal elements of $\\e{M}$) are chosen in such a way, that it is beneficial to copy (share) the data and then transmit together, instead of broadcasting the data directly. \nSpecifically, we set $\\e{m}^1 = \\frac{1}{4}$ and $\\e{m}^2 = \\e{m}^3 = 1$ and assume all links to be disjunct. Note, that we can neglect $q^3$ in all further discussions, since it only symbolizes the destination queue.\n\nFor this simple example, it is prudent to forgo working in terms of the \\textit{control vector} $v_t$ and instead use the \\textit{control option} $u_t$, which we define to be $u_t = R \\e{M} v_t$ (this $u_t$ is not related to the one, that was used in the definition of the $\\mathsf{PNC}$ policy). We have $u_t \\in \\mathcal{U}$, which can be derived from the system description without further ado to be\n\\begin{equation}\n    \\mathcal{U} =\n    \\Set{\n    u^0 , u^1 , u^2 , u^3\n    }\n    = \n    \\Set{\n    \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} ,\n    \\begin{pmatrix} -1 \\\\ 0 \\end{pmatrix} ,\n    \\begin{pmatrix} 0 \\\\ 4 \\end{pmatrix} ,\n    \\begin{pmatrix} -4 \\\\ -4 \\end{pmatrix}\n    }\n\\end{equation}\nwhere we scaled all elements of $\\mathcal{U}$ with the factor $4$ to simplify any calculations. We have $u^1, u^2, u^3$ represent single transmission, data sharing, and joint transmission, respectively and in each time slot, the controller may only choose one of these options to influence the expected queue state via $\\CE{q_{t+1}}{q_t} = q_t + u_t + a_t$.\n\nRegarding \\eqref{eq::def_to}, it is now very easy to express the set of all arrival rates $\\e{a}$, for which there exists a policy that stabilizes the system. We call this set the maximum stability region $\\mathcal{A}$ and have\n\\begin{equation}\n\t\\begin{aligned}\n\t\\mathcal{A} :&= \\Set{\n\t\t\\e{a} : \\quad\n\t\t\\e{a} + \\sum_{u \\in \\mathcal{U}} \\lambda^u u = -\\mathbf{1} \\varepsilon, \\quad\n\t\t\\begin{gathered}\n\t\t\t\\varepsilon > 0\n\t\t\t\\\\\n\t\t\t\\sum \\lambda^u \\leq 1\n\t\t\\end{gathered}\n\t}\n\t\\\\\n\t&=\n\t\\Set{\n\t\t\\e{a} : \\quad\n\t\t\\e{a} + \\sum_{u \\in \\mathcal{U}} \\lambda^u u =  \\mathbf{0}, \\quad \\hspace{4.5mm}\n\t\t\t\\sum \\lambda^u < 1\n\t}\n\t\\end{aligned}\n\\end{equation}\nRemember that throughput optimality is accomplished, if a policy can stabilize the system for all arrival rates in $\\mathcal{A}$.\nA graphical representation of $\\mathcal{A}$ is given in green on the left side of \\figref{fig::constructive_interference_stab_region}.\n\nNow, let us assume that there is no arrival at $q^2$, i.e. $\\e{a}^2 = 0$.\nUsing the control options $u^2$ and $u^3$ in alternating sequence (given that there are enough packets in $q^1$ to do so) would yield an efflux of $4$ packets every $2$ time slot, thus an efflux of $2$ packets per time slot. The corresponding point is shown on the right side of \\figref{fig::constructive_interference_stab_region}.\nIt is easy to check, that no other sequence of control options can match this efflux.\n\nHowever, conventional back-pressure policies like $\\mathsf{MW}$ are not able to access the control option $u^2$, resulting in the loss of its throughput optimality in this example. The only arrival rates, that $\\mathsf{MW}$ \\textit{can} stabilize are those in the red triangle on the right side of \\figref{fig::constructive_interference_stab_region}.\n\n\\begin{figure}[htbp]\n  \\centering\n \n  \\includegraphics[]{stab_region_simple_f.pdf}\n  \\vspace{0mm}\n    \\centering\n  \\caption{Example 2 - Stability Regions}\n  \\label{fig::constructive_interference_stab_region}\n\\end{figure}\n\nIn contrast, $\\mathsf{PNC}$ is able to select the missing control option $u^2$ and simulations suggest, that it stabilizes the example for any strictly positive arrival rate $\\e{a}$ from $\\mathcal{A}$. To substantiate this claim we refer to \\figref{fig::simu_comparison}. Here, we simulated the queue state $q^1$ over time $t$ for 3 different arrival rates $\\e{a}$ under 3 different policies. The respective positions of those $\\e{a}$ regarding $\\mathcal{A}$ are depicted in \\figref{fig::simu_points}. As for the policies, we chose $\\mathsf{MW}$ and $\\mathsf{PNC}$. Also, we added a third control policy, labeled $\\mathsf{fPNC}$ for \\textit{fixed} $\\mathsf{PNC}$. This policy mimics the $\\mathsf{PNC}$ policy, except that it uses the entire calculated control trajectory before repeating the optimization. In contrast, $\\mathsf{PNC}$ repeats the optimization every time slot again.\n\nAs predicted, we have $\\mathsf{MW}$ not stabilizing the blue and green arrival rates. Furthermore, it can be seen, that $\\mathsf{fPNC}$ loses some stabilizing properties with increasing horizon as the green arrival rate cannot be stabilized with $H=3$ (this is related to the horizon not being an even number). This proves, that the MPC paradigm of repeating the optimization in every step (and thereby discarding the rest of the trajectory) is an essential part in the $\\mathsf{PNC}$ policy.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{stab_region_points_f.pdf}\n  \\caption{Example 2 - Selected arrival rates for simulation; Corresponding stability regions: (red -- $\\mathsf{MW}$), (red+blue -- $\\mathsf{fPNC}_{H=3}$), (red+blue+green -- $\\mathsf{PNC}$, $\\mathsf{fPNC}_{H=2}$)}\n  \\label{fig::simu_points}\n\\end{figure}\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[]{horizon_dependent_stability_f.pdf}\n  \\caption{Example 2 - \n  Queue state (system state) $q^1$ as a function of time $t$ for various (color-coded) arrival rates}\n  \\label{fig::simu_comparison}\n\\end{figure}\n\n\\section{Conclusion}\nWe successfully modify a discrete-time queueing network with a JMS, i.e. with an additional DTMC that changes network parameters (even topology) on a mid- to long-term time scale. We then introduce a novel family of predictive control policies, $\\mathsf{PNC}$, based on the paradigms of MPC, and devise a special implementation of the underlying prediction, that allows the policy to be executed in the fastest way possible. The policy is especially well suited to control the mentioned systems and outperforms conventional control approaches as is illustrated in numerical simulations. In our main contribution, we prove throughput optimality of $\\mathsf{PNC}$. Looking ahead, we see an intriguing application in networks that consist of synchronized queues (e.g. found in parallel computing or manufacturing chains). Those networks still elude known control strategies but seem to be stabilizable under $\\mathsf{PNC}$ policies with suitably chosen prediction horizon.\n\n\\section*{Acknowledgment}\nThis work is part of and thereby funded by the DFG Priority Program 1914\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation} \n Information on hyperon resonances is generally not as extensive as for nucleon resonances. The study of $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$ could lead to the better understanding of $\\Lambda^*$s and $\\Sigma^*$s.\n\n     Most previous partial-wave analyses (PWAs) of $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$ {\\cite{Armenteros1969, Conforto1971, Horn1975_1, Hemingway1975, Baillon1975,  Gopal1977}}, were based on the assumption that  partial-wave amplitudes could be represented by a simple sum of resonant and  background terms. Such an assumption violates  unitarity of the partial-wave $S$-matrix. In this work, we report on our investigation of the reactions $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, $K^-p\\rightarrow \\pi^0\\Lambda$, and $K^-p\\rightarrow\\pi^+\\Sigma^-$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$ via single-energy analyses and a subsequent energy-dependent analysis.  All available differential cross section, polarization, polarized cross section, and cross-section data up to a maximum c.m.\\ energy of about 2.1 GeV were fitted. In order to ensure that our amplitudes had a relatively smooth variation with energy, we introduced several constraints that will be described in detail below. The determination of resonance parameters from our subsequent energy-dependent analysis is discussed in Ref. \\cite{manoj2013}.\n\n\n         \n \n\n   \n   \\section{Formalism and Fitting Procedures}\n        Here, we summarize the formalism for the single-energy partial-wave analyses.\n        The differential cross section $\\rm d\\sigma \/\\rm d\\Omega$ and polarization $P$ for unpolarized scattering of spin-0 mesons off spin-$\\frac12$ nucleons are given by \\cite{bransden73}\n               \\begin{equation}\n               \\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega} = {\\lambdabar}^2(|f|^2+|g|^2)~,\n                \\end{equation}\n                \\begin{equation}\n                P\\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega} =2{\\lambdabar}^2\\rm Im(fg^\\ast)~,\n                \\end{equation}\n        where $\\lambdabar = {\\hbar}\/{k}$,\n         with $k$ the magnitude of c.m.\\ momentum of the incoming meson.\n         Here, $f = f(W,\\theta)$ and $g = g(W,\\theta)$ are the usual spin-non-flip and spin-flip amplitudes at c.m.\\ energy $W $ and meson c.m.\\ scattering angle $\\theta$. In terms of partial waves, $f$ and $g$ can be expanded as\n       \n       \\begin{equation}\n        f(W,\\theta) = \\sum_{l=0}^{\\infty} [(l+1)T_{l+} + lT_{l-}]P_l(\\cos\\theta)~,\n        \\end{equation}\n        \\begin{equation}\n        g(W,\\theta) = \\sum_{l=1}^{\\infty} [T_{l+} - T_{l-}]P_l^1(\\cos\\theta)~,\n        \\end{equation}\n        where $l$ is the initial orbital angular momentum, $P_l(\\cos\\theta)$ is a Legendre polynomial and $P_l^1(\\cos\\theta) = \\sin\\theta \\cdot {\\rm d} P_l(\\cos\\theta)\/{\\rm d}(\\cos\\theta$). The total angular momentum for the amplitude $T_{l+}$ is $J=l+\\frac12$, while that for the amplitude $T_{l-}$ is $J=l-\\frac12$.\n              For the initial $\\overline K N$ system, we have $I = 0$ or $I =1$ so that the amplitudes $T_{l\\pm}$ can be expanded in terms of  isospin amplitudes as \n         \\begin{equation}\n               T_{l\\pm} = C_{0}T^{0}_{l\\pm} + C_{1}T^{1}_{l\\pm}~,\n       \\end{equation}\n       \\newline\n       where $T^I_{l\\pm}$ are partial-wave amplitudes\n        with isospin $I$  and total angular momentum $J = l\\pm\\frac12$ with $C_I$ the appropriate isospin Clebsch-Gordon coefficients for a given reaction.\n       For $K^- p \\rightarrow K^- p$, for example, we have $C_{0} = {\\frac12}$ and $C_{1}={\\frac12}$.\n\\newline\n\n        The total $K^- p$ cross section is given by $\\sigma_{\\rm total} = 4\\pi {\\lambdabar}^2 {\\rm Im} f(W, 0)$, or\n\\begin{equation}\\label{eq:Sigma_total}\n\\sigma_{\\rm total} = 4\\pi \\lambdabar^2 \\sum_{l=0}^{\\infty}[(l+1) {\\rm Im}T_{l^+} + l{ \\rm Im}T_{l^-}],\n\\end{equation}\nwhere here the $T_{l\\pm}$ are partial-wave amplitudes for elastic $\\overline KN$ scattering.\n\n\n  The integrated cross section  for a particular two-body reaction is\n\\begin{equation}\n\\sigma=4\\pi {\\lambdabar}^2 \\sum_{l=0}^{\\infty} [(l+1) |T_{l^+}|^2+l |T_{l^-}|^2] .\n\\end{equation}\n\n  Tables I, II, and III summarize  the available quantity and types of data in each energy bin for the three reactions $\\overline K N\\rightarrow \\overline K N$, $\\overline K N\\rightarrow \\pi \\Lambda$, and $\\overline K N\\rightarrow \\pi \\Sigma$, respectively.   \n   Single-energy fits were performed separately for (i)  $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, for (ii) $K^-p\\rightarrow \\pi^0\\Lambda$, and for (iii) $K^-p\\rightarrow\\pi^+\\Sigma^-$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$. In each case the available data were analyzed in  c.m.\\ energy bins of width 20 MeV. This choice of bin width was appropriate because the data for smaller widths had unacceptably low  statistics and for larger  widths, some amplitudes varied too much over the energy spread of the bin. \n \n\n\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\overline{K} N$. Column 1 lists the central energy $W_0$ of each energy bin, columns 2 and 3 list the number of differential cross-section data points in each bin for $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$, respectively, column 4 lists the number of polarization data points for $K^-p\\rightarrow K^-p$ in each bin, column 5 lists the number of polarized cross-section data points for $K^-p\\rightarrow K^-p$ in each bin, column 6 lists the number of integrated cross-section data points for $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$ in each bin, column 7 lists the number of $K^-p$ total cross-section data points in each bin, column 8 lists the total number of data points for all kinds of data in each bin, and column 9 lists the references for the measurements referred to in columns 2-5.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccc}\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$ } } & \\multicolumn{1}{c}{${d\\sigma}\/{d\\Omega}$ { } }& \\multicolumn{1}{c}{{\\mbox{$P$}}}& \\multicolumn{1}{c}{{\\mbox{$P$}${d\\sigma}\/{d\\Omega}$ }} & \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma$}}& \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma_{\\rm total}$}} & \\multicolumn{1}{c}{ Total} & \\multicolumn{1}{c}{\\multirow{2}{*}{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} & ($K^- p$) &($\\overline{K}^0 n$) & ($K^- p$)& ($K^- p$)& & & \\multicolumn{1}{c}{ No.} &  \\multicolumn{1}{c}{} \\\\\n\n\\hline\n1480\t&\t 57&\t 24&\t&\t\t&15\t&\t5&   96\t&\t \\cite{Mast1976}\\\\\n1500\t&\t114&\t120&\t&&\t\t23\t&\t8&  257\t\t& \\cite{Mast1976}\\\\\n1520\t&\t100&\t100&\t&&\t\t28\t&  10&  228\t\t& \\cite{Mast1976}\\\\\n1540\t&\t178&\t120&\t&&\t\t25\t&\t8&  323\t\t& \\cite{Armenteros1970, Mast1976}\\\\\n1560&\t117&\t100&\t&&\t\t14\t&\t5&  231\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1580&\t78&\t112&\t&\t&\t10\t&\t3&   200\t\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1600&\t79&\t116&\t&\t&\t14\t&\t6&   209\t\t\t& \\cite{Armenteros1970, Alston1978_1, CrystalBall2005}\\\\\n1620&\t147&\t120&\t&&\t\t14\t&\t6&  281\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1640\t&\t113&\t148&\t&&\t\t14\t&\t6&  275\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1660\t&\t149&\t132&\t&&\t\t19\t&\t6&  300\t\t& \\cite{Armenteros1970, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1680\t&\t194&\t210&\t&&\t\t30\t&\t9&  434\t\t& \\cite{Armenteros1968, Armenteros1970, Conforto1971, Adams1975, Alston1978_1, CrystalBall2005}\\\\\n1700\t&\t150&\t112&\t&&\t\t11\t&\t6&  283\t\t& \\cite{Armenteros1968, Armenteros1970, Conforto1971, Adams1975, Alston1978_1}\\\\\n1720\t&\t150&\t150&\t&&\t\t20\t&\t6&  320\t\t& \\cite{Armenteros1968, Conforto1971, Adams1975, Alston1978_1}\\\\\n1740\t&\t216&\t241&\t&27&\t\t29\t&\t6&    513\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Jones1975, Adams1975, Alston1978_1}\\\\\n1760&\t176&\t193&\t&26&\t\t25\t&\t4&    420\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Jones1975, Adams1975, Alston1978_1}\\\\\n1780\t&\t185&\t196&\t&27&\t\t34\t&\t9&    442\t\t & \\cite{Armenteros1968, Andersson1970, Conforto1971, Jones1975, Conforto1976, Alston1978_1}\\\\\n1800&\t132&\t79&\t&52&\t\t18\t&\t4& 281\t\t\t& \\cite{Armenteros1968, Conforto1971, Jones1975, Conforto1976}\\\\\n1820&\t146&\t60&\t&27&\t\t18\t&\t5& 251\t\t\t& \\cite{Armenteros1968, Albrow1971, Conforto1971, Conforto1976}\\\\\n1840&\t185&\t80&\t&27&\t\t23\t&\t9& 315\t\t\t& \\cite{Armenteros1968, Andersson1970, Conforto1971, Conforto1976}\\\\\n1860&\t227&\t100&\t&30&\t\t22\t&\t4&    379\t\t & \\cite{Armenteros1968, Andersson1970, Conforto1971, Griselin1975, Conforto1976}\\\\\n1880&\t266&\t120&\t&28&\t\t21\t&\t6&    435\t\t & \\cite{Armenteros1968, Albrow1971, Conforto1971, Griselin1975, Conforto1976}\\\\\n1900\t&\t175&\t60&\t&56&\t\t18\t&\t6& 309\t\t\t& \\cite{Armenteros1968, Andersson1970, Albrow1971, Conforto1971, Griselin1975, Conforto1976}\\\\\n1920\t&\t146&\t60&\t&27&\t\t17\t&\t3& 250\t\t\t& \\cite{Andersson1970, Griselin1975, Conforto1976}\\\\\n1940&\t110&\t80&\t&30&\t\t18\t&\t5& 238\t\t\t& \\cite{Albrow1971, Griselin1975, Conforto1976}\\\\\n1960&\t64&\t40&\t23&&\t\t14\t&\t4&  141\t\t& \\cite{Daum1968, Griselin1975, Conforto1976}\\\\\n1980&\t34&\t20&\t&&\t\t11\t&\t3&    65\t\t& \\cite{Griselin1975, Abe1975}\\\\\n2000&\t23&\t20&\t23&&\t\t9\t&\t3&   75\t\t& \\cite{Daum1968, Abe1975}\\\\\n2020\t&\t23&\t&\t23&&\t\t12\t&\t5&    58\t\t& \\cite{Daum1968}\\\\\n2040\t&\t54&\t&\t22&&\t\t10\t&\t4&    86\t\t& \\cite{Daum1968, Abe1975}\\\\\n2060&\t23&\t&\t23&&\t\t10\t&\t3&    56\t\t& \\cite{Daum1968}\\\\\n2080\t&\t22&\t&\t22&&\t\t9\t&\t3& 53\t\t& \\cite{Daum1968}\\\\\n2100&\t53&\t&\t22&&\t\t7\t&\t2& 83\t\t& \\cite{Daum1968, Abe1975}\\\\\n2120\t&\t46&\t&\t23&46&\t\t12\t&\t5&    104\t\t& \\cite{Daum1968, Andersson1970}\\\\\n2140&\t23&\t&\t23&&\t\t5\t&\t2& 51\t\t& \\cite{Daum1968}\\\\\n2160&\t32&\t&\t&&\t\t14\t&\t5&  46\t\t& \\cite{Abe1975}\\\\\n\\end{tabular}\n\\label{Table:KN}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\pi \\Lambda$. Column 1 lists the central energy $W_0$ of each energy bin, column 2 lists the number of differential cross-section data points in each bin for $K^-p\\rightarrow \\pi^0\\Lambda$, column 3 lists the number of polarization data points in each bin, column 4 lists the number of polarized cross-section data points in each bin, column 5 lists the number of integrated cross-section data points in each bin, column 6 lists the total number of data points for all kinds of data in each bin, and column 7 lists the references for the measurements referred to in columns 2-4.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{\\multirow{2}{*}{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{\\multirow{2}{*}{\\mbox{$P$}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{$\\sigma$}} & \\multicolumn{1}{c}{ Total}  & \\multicolumn{1}{c}{\\multirow{2}{*}{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} & & & & & \\multicolumn{1}{c}{ No.} & \\multicolumn{1}{c}{} \\\\\n\\hline\n1480\t&\t\t&  &\t\t&\t4\t&\t4\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1500&\t\t&  &\t\t&\t8\t&\t8\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1520\t&\t\t&  &\t\t&\t9\t&\t9\t\t\t&\t\t \\cite{Baldini1988}\\\\\n1540&\t40\t&  &\t16\t&\t7\t&\t63\t\t\t&\t\t \\cite{Armenteros1970}\\\\\n1560&\t76\t&16&\t26\t&\t4\t&\t122\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1580&\t56\t&16&\t19\t&\t2\t&\t93\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1600\t&\t56\t&16&\t20\t&\t5\t&\t97\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1620\t&\t56\t&16&\t20\t&\t4\t&\t96\t\t\t&\t \\cite{Armenteros1970, CrystalBall2005}\\\\\n1640&\t81\t&32&\t20\t&\t6\t&\t139\t\t\t&\t \\cite{Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1660\t&\t74\t&16&\t19\t&\t9\t&\t118\t\t\t&\t \\cite{Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1680\t&\t114\t&22&\t28\t&\t13\t&\t177\t\t\t&\t \\cite{Armenteros1968, Armenteros1970, Baxter1973, CrystalBall2005}\\\\\n1700\t&\t58\t&7 &\t\t8\t&\t13\t&\t86\t\t\t\t&\\cite{Armenteros1968, Armenteros1970, Baxter1973}\\\\\n1720\t&\t101\t&27&\t\t&\t9\t&\t137\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1740&\t185\t&65&\t\t&\t12\t&\t262\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1760\t&\t138\t&54&\t\t&\t11\t&\t203\t\t\t&\t \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1780&\t160\t&88&\t\t&\t10\t&\t258\t\t\t&\t \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1800\t&\t80\t&46&\t\t&\t4\t&\t130\t\t\t&\t \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1820\t&\t60\t&35&\t\t&\t4\t&\t99\t\t\t&\t \\cite{Armenteros1968, Conforto1976}\\\\\n1840&\t80\t&36&\t\t&\t5\t&\t121\t\t\t&\t \\cite{Armenteros1968, Conforto1976}\\\\\n1860&\t100\t&26&\t\t&\t7\t&\t133\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1880\t&\t120\t&44&\t\t&\t7\t&\t171\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1900\t&\t60\t&18&\t\t&\t5\t&\t83\t\t\t&\t \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1920\t&\t80\t&21&\t\t&\t7\t&\t108\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1940\t&\t100\t&23&\t\t&\t8\t&\t131\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1960\t&\t60\t&24&\t\t&\t5\t&\t89\t\t\t&\t \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1980&\t40\t&6&\t\t&\t5\t&\t51\t&\t\t\t \\cite{Berthon1970, Griselin1975}\\\\\n2000\t&\t40\t&13&\t&\t4\t&\t57\t&\t\t\t \\cite{Berthon1970, Griselin1975}\\\\\n2020\t&\t20\t&9&\t\t&\t4\t&\t33\t\t&\t \\cite{Berthon1970}\\\\\n2040\t&\t20\t&8&\t\t&\t2\t&\t30\t\t&\t \\cite{Berthon1970}\\\\\n2060&\t30\t&6&\t\t&\t3\t&\t39\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2080&\t40\t&8&\t\t&\t3\t&\t51\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2100&\t30\t&&\t\t&\t2\t&\t32\t\t&\t \\cite{London1975}\\\\\n2120\t&\t70\t&11&\t&\t4\t&\t85\t\t&\t \\cite{Berthon1970, London1975}\\\\\n2140&\t\t& &\t\t&\t\t&\t0\t   \t &\tNA\\\\\n2160\t&\t40\t&9&\t\t&\t2\t&\t51\t\t&\t \\cite{Berthon1970}\\\\\n\\end{tabular}\n\\label{Table:Lambda}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}[htbp]\n\\caption{Summary of database for $\\overline{K} N \\rightarrow \\pi \\Sigma$. Column 1 lists the central energy $W_0$ of each energy bin, columns 2, 3, and 4 list the number of differential cross-section data points in each bin for $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, column 5 and 6 list the number of polarization data points for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, in each bin, column 7 and 8 list the number of polarized cross-section data points for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively, in each bin, column 9 lists the number of integrated cross-section data points for $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\pi^-\\Sigma^+$ in each bin, column 10 lists the total number of data points for all kinds of data in each bin, and column 11 lists the references for the measurements referred to in columns 2-8.}\n\\begin{center}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc\n\\multicolumn{1}{c}{$W_{0}$} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{${d\\sigma}\/{d\\Omega}$}} & \\multicolumn{1}{c}{{\\mbox{$P$}}}& \\multicolumn{1}{c}{{\\mbox{$P$}}} & \\multicolumn{1}{c}{{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}}& \\multicolumn{1}{c}{{\\mbox{{$P$}${d\\sigma}\/{d\\Omega}$}}} & \\multicolumn{1}{c}{{$\\sigma$}} & \\multicolumn{1}{c}{ Total}  & \\multicolumn{1}{c}{{ References}} \\\\\n\\multicolumn{1}{c}{\\rm (MeV)} &{($\\pi^+ \\Sigma^-$})  &{($\\pi^0 \\Sigma^0$}) & {($\\pi^- \\Sigma^+$}) & {($\\pi^0\\Sigma^0$})& {($\\pi^- \\Sigma^+$})& {($\\pi^0\\Sigma^0$})& {($\\pi^- \\Sigma^+$})& & \\multicolumn{1}{c}{ No.} &  \\multicolumn{1}{c}{} \\\\\n\\hline\n1480&\t\t&\t\t&\t\t&&&\t\t& &\t13\t&\t13\t\t&\t \\cite{Baldini1988}\\\\\n1500\t&\t\t&\t\t&\t\t&&&\t\t&\t&21\t&\t21\t\t&\t \\cite{Baldini1988}\\\\\n1520\t&\t\t&\t\t&\t\t&&&\t\t&\t&23\t&\t23\t\t&\t \\cite{Baldini1988}\\\\\n1540\t&\t40\t&\t19\t&\t40\t&&&\t19\t&18\t&22\t&\t158\t\t&\t  \\cite{Armenteros1970}\\\\\n1560\t&\t60\t&\t39\t&\t60\t&9&&\t30\t&30&\t12\t&\t240\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1580&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&6\t&\t164\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1600&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&10\t&\t168\t\t&\t  \\cite{Armenteros1970, CrystalBall2005, CrystalBall2008_1}\\\\\n1620\t&\t40\t&\t29\t&\t40\t&9&\t&20\t&20\t&13\t&\t171\t\t&\t  \\cite{Armenteros1970,CrystalBall2005, CrystalBall2008_1}\\\\\n1640\t&\t40\t&\t48\t&\t40\t&18&&\t20\t&20&\t11\t&\t\t&\t  \\cite{Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1660\t&\t40\t&\t49\t&\t40\t&9&\t&20\t&\t19&17\t&\t194\t\t&\t  \\cite{Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1680\t&\t80\t&\t49\t&\t80\t&9&\t&30\t&29\t&25\t&\t302\t\t&\t  \\cite{Armenteros1968, Armenteros1970, Baxter1973, CrystalBall2005, CrystalBall2008_1}\\\\\n1700\t&\t40\t&\t30\t&\t40\t&&\t&10\t&9\t&16\t&\t145\t\t&\t  \\cite{Armenteros1968, Armenteros1970, Baxter1973}\\\\\n1720&\t76\t&\t30\t&\t75\t&&10&\t\t&&\t15\t&\t206\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1740&\t154\t&\t20\t&\t157\t&&24&\t\t&&\t24\t&\t379\t\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1760\t&\t114\t&\t20\t&\t114\t&&25&\t\t&&\t20\t&\t293\t\t&\t  \\cite{Armenteros1968, Baxter1973, Jones1975}\\\\\n1780\t&\t148\t&\t\t&\t147\t&&34&\t\t&&\t24\t&\t353\t\t&\t  \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1800\t&\t72\t&\t\t&\t74\t&&22&\t\t&&\t10\t&\t178\t\t&\t  \\cite{Armenteros1968, Jones1975, Conforto1976}\\\\\n1820&\t60\t&\t\t&\t60\t&&15&\t\t&&\t11\t&\t146\t\t&\t  \\cite{Armenteros1968, Conforto1976}\\\\\n1840\t&\t80\t&\t\t&\t80\t&&13&\t\t&&\t14\t&\t187\t\t&\t  \\cite{Armenteros1968, Conforto1976}\\\\\n1860&\t100\t&\t\t&\t100\t&&11&\t\t&&\t18\t&\t229\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1880&\t120\t&\t\t&\t120\t&&24&\t\t&&\t18\t&\t282\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1900&\t60\t&\t\t&\t60\t&&14&\t\t&&\t8\t&\t142\t\t&\t  \\cite{Armenteros1968, Griselin1975, Conforto1976}\\\\\n1920\t&\t80\t&\t\t&\t79\t&&15&\t\t&&\t12\t&\t186\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1940\t&\t99\t&\t\t&\t100\t&&14&\t\t&&\t17\t&\t230\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1960\t&\t60\t&\t\t&\t59\t&&17&\t\t&&\t10\t&\t146\t\t&\t  \\cite{Berthon1970, Griselin1975, Conforto1976}\\\\\n1980\t&\t40\t&\t\t&\t38\t&&\t&\t\t&\t&7\t&\t85\t\t&\t  \\cite{Berthon1970, Griselin1975}\\\\\n2000\t&\t40\t&\t\t&\t40\t&&\t&\t&\t&9\t&\t89\t&\t\t  \\cite{Berthon1970, Griselin1975}\\\\\n2020\t&\t19\t&\t\t&\t19\t&&\t&\t&\t&5\t&\t43\t\t&\t  \\cite{Berthon1970}\\\\\n2040&\t20\t&\t\t&\t16\t&&\t&\t&\t&2\t&\t38\t\t&\t  \\cite{Berthon1970}\\\\\n2060\t&\t19\t&\t10\t&\t18\t&&\t&\t&\t&5\t&\t52\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2080\t&\t20\t&\t10\t&\t19\t&&\t&\t&\t&5\t&\t54\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2100\t&\t\t&\t26\t&\t\t&&\t&\t&\t&3\t&\t29\t\t&\t  \\cite{London1975}\\\\\n2120\t&\t39\t&\t8\t&\t37\t&&\t&\t&\t&4\t&\t88\t\t&\t  \\cite{Berthon1970, London1975}\\\\\n2140\t&\t\t&\t\t&\t\t&&\t&\t&\t\t&\t0       \t\t    &\t  NA\\\\\n2160&\t36\t&\t\t&\t35\t&&\t&\t&\t&5\t&\t76\t\t&\t  \\cite{Berthon1970}\\\\\n\\end{tabular}\n\\label{Table:Sigma}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n \n \n The general qualitative behavior of the partial-wave amplitudes that we wanted to determine is known from earlier partial-wave analyses. Therefore it was convenient to make use of this information in our single-energy fits. In 2007, one of us (J. Tulpan) completed a multichannel fit {\\cite{Tulpan2007}} of published partial-wave amplitudes for $\\overline{K}N$ scattering to several final states, including $\\overline{K}N$, $\\overline{K}^*N$, $\\overline{K}\\Delta$, $\\pi \\Lambda$, $\\pi \\Lambda(1520)$, $\\pi \\Sigma$, and $\\pi \\Sigma(1385)$. The channels $\\sigma\\Lambda$, $\\sigma\\Sigma$, and $\\eta\\Sigma$ (for $S_{11}$) were included as ``dummy channels\",  where $\\sigma$ denotes the broad isoscalar S-wave $\\pi \\pi$ interaction. Also, $\\eta\\Lambda$ was included for $S_{01}$. Our fit of $S_{01}$ amplitudes included data for $\\sigma(K^-p\\rightarrow\\eta\\Lambda)$ up to a c.m.\\ energy of 1685 MeV (see Fig.\\ \\ref{etalambda}). The dummy channels were channels without data and were included to satisfy unitarity in some partial waves.\nTulpan's work resulted in an energy-dependent solution that is consistent with { $S$}-matrix unitarity.\nWe refer to his solution as the {\\it initial global fit}. Within each energy bin, each partial-wave amplitude with a given isospin amplitude was approximated by a first-order Taylor series expansion:\n\\begin{equation}\n       T(W) = T(W_0) + T'(W_0)(W - W_0)\n\\end{equation}       \n        where $W$ is the c.m.\\ energy of the data point in the bin and $W_0$ is the central energy of the bin. Here, for simplicity $T(W)$ denotes an isospin amplitude $T_{l\\pm}^I$. The complex $T$-matrix amplitude $T(W_0)$ belongs to the parameter set to be optimized at c.m.\\ energy $W_0$, and $T'(W_0)$ is called the slope parameter. During fits, the slope parameter was held fixed so that the real and imaginary parts of $T(W_0)$ were our fitting parameters.\nDuring our initial single-energy partial-wave analyses, we calculated the slope parameters $T'(W_0)$ from the initial global fit and kept these parameters constant in our fits.\n\n\\begin{figure}[htpb]\n\\scalebox{0.35}{\\includegraphics{.\/etalambda.pdf}}\n\\caption{Integrated cross section for $K^-p\\rightarrow\\eta\\Lambda$ compared with the results of our energy-dependent fit. Data are from Starostin 2001 \\cite{starostin2001}.}\n\\label{etalambda}\n\\end{figure}\n\nBecause the database is somewhat sparse, additional constraints were introduced in order to determine partial-wave amplitudes with a reasonably smooth variation with energy.\nTo decrease the number of free parameters to be searched, we also held fixed the very small {$ T$}-matrix amplitudes (those with $|{ T}(W_0)|<0.05$).\nThis constraint is expected to introduce only a small bias to our final energy-dependent partial-wave solution.\n\nAs an additional constraint, we held fixed the ${ D}_{03}$ amplitudes for $\\overline{K}N \\rightarrow \\overline{K}N$ and $\\overline{K}N \\rightarrow \\pi \\Sigma$ at the values from the initial global fit in the bin with $W_0=1520$ MeV. This constraint was introduced because of the well-known narrow $\\Lambda (1520)$ resonance, which has a width of only about 16 MeV.\nEven with this constraint, we ultimately concluded that we could not determine reliable amplitudes in this bin for the  reactions $\\overline{K}N \\rightarrow \\overline{K}N$ and $\\overline{K}N \\rightarrow \\pi \\Sigma$.\n\nFinally, in our single-energy fits, we introduced a {\\it penalty term} to the $\\chi^2$ function that we minimized. This penalty term constrained our fitted amplitudes from differing greatly from the values of the initial global fit. \nFor calculating the uncertainties in our single-energy amplitudes, we carried out a {\\it zero-iteration} fit in which the initial values of all amplitudes were replaced by the values determined by our $\\chi^2$ minimization procedure.\nIn the zero-iteration fit, all partial-wave amplitudes except ${ G}_{17}$ were treated as free parameters.\nThe ${ G}_{17}$ amplitude was held fixed in this procedure to remove the ambiguity in determining the global overall phase of our amplitudes. Spin-9\/2 waves were not needed in our solution.\n\nOnce we had obtained a complete set of amplitudes for $\\overline{K}N$, $\\pi \\Lambda$, and $\\pi \\Sigma$ reactions from our single-energy analyses, we carried out  global multichannel energy-dependent fits using a procedure similar to that of Tulpan \\cite{Tulpan2007}.\nThe key new ingredient is that our global fit (details of how the partial-wave $S$-matrix was constructed can be found in Ref. \\cite{manoj12}) used our own single-energy amplitudes for the $\\overline{K}N$, $\\pi \\Lambda$, and $\\pi \\Sigma$ channels.\nFor other final states,\nwe used the same input information as Tulpan did from Refs. \\cite{Gopal1977, Cameron1978, Cameron1977, Cameron1978_1, Litch1974}. We assumed the same uncertainties used by Tulpan {\\cite{Tulpan2007}} for obtaining the inital global fit ($\\pm 0.025$ for $\\overline{K}N$, $\\pm 0.035$ for $\\pi \\Lambda$ and $\\pi \\Sigma$, and $\\pm 0.050$ otherwise). These uncertainties were necessary because previous published partial-wave amplitudes were without error bars. These uncertainties were estimated by comparing like partial-wave amplitudes from different energy-dependent analyses and estimating the average differences for the real and imaginary parts. The smaller error bars implied the analyses agreed well with each other and the larger error bars implied the analyses agreed less well with each other.\n\nWe reduced the number of free amplitudes for a new set of single-energy solutions. At this stage, our free amplitudes included only $ S_{01}, S_{11}, P_{01}, P_{11}, P_{13},$ and $ D_{03}$.\nAll other amplitudes were held fixed at the values determined from our first new global fit.\nIn addition, the slope parameters were recalculated based on the new global fit and kept constant during this stage of the single-energy analyses.\nWe were able to obtain excellent agreement with the observables.\nNext, we repeated our global energy-dependent analysis to refit the new set of single-energy amplitudes for $ S_{01}, S_{11}, P_{01}, P_{11}, P_{13},$ and $ D_{03}$.\nWe then compared our new predictions with the observables in our single-energy fits.\nStill the agreement was less than satisfactory, so we carried out yet another round of single-energy analyses.\nAt this stage, we successfully reduced our free amplitudes to include only $ S_{01}, S_{11}$, and $ P_{01}$.\nAll other amplitudes were unchanged at the values from our last global fit, and slope parameters were again recalculated from the last global fit, and then held fixed in the single-energy fits.\n\n\\section{\\emph{\\bf  RESULTS AND DISCUSSION }}\nThe final single-energy fits resulted in an excellent agreement with all observables ($\\rm d\\sigma\/{\\rm d\\Omega}$, $P$, $P\\rm d\\sigma\/{\\rm d\\Omega}$, and $\\sigma$) yielding a fairly smooth set of partial-wave  amplitudes within the energy range of our analysis. The energy-dependent solutions were finally used to compare with the observable data.\nFigures 2 - 7 show representative energy-dependent results for the differential cross section of each $\\overline KN$ reaction included in our single-energy fits. \nThe cross sections are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson.\nFigure 2 shows the comparison of differential cross section data for $K^-p\\rightarrow K^-p$ with our energy-dependent solution at four lab momenta of 514, 935, 1165, and 1483 MeV. Although  the data are from the 1960s and 1070s \\cite{Armenteros1970, Conforto1971, Conforto1976, Daum1968} they are in excellent agreement with our solution. For $K^- p \\rightarrow \\overline K^0 n$ (Fig.\\ 3) the Crystal Ball data with smaller error bars at $ P_{\\rm Lab}$ = 514 MeV and 714 MeV are well described by our solution in the forward and backward angles with a slight under fitting in the intermediate angles. The other data \\cite{Jones1975, Griselin1975} at $P_{\\rm Lab}$ = 936 MeV and 1434 MeV with larger error bars are in good agreement with our energy-dependent solution. Similarly, Fig.\\ 4 shows the excellent agreement between our solution and differential cross section data at $P_{\\rm Lab}$ = 514, 750, 1153, and 1465 MeV for $K^-p\\rightarrow \\pi^0\\Lambda$. Figure 5 shows a comparison of data from Refs. \\cite{Armenteros1970, Armenteros1968, Conforto1976, Berthon1970} with our solution for $K^-p\\rightarrow \\pi^+\\Sigma^-$. Except for some under representation of data at $P_{\\rm Lab}$ = 1245 MeV we have an excellent agreement with the data. Figures 6 and 7 show an excellent agreement of our energy-dependent solution with the differential cross section data at various lab momenta of kaons for $K^-p\\rightarrow \\pi^0\\Sigma^0$ and $K^-p\\rightarrow \\pi^-\\Sigma^+$, respectively.\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_935.pdf}}\n\\vspace{-25.5mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_1165.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_11_1483.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Conforto 1971 \\cite{Conforto1971}, and Duam 1968 \\cite{Daum1968}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_750.pdf}}\n\\vspace{-25.5mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_1165.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_12_1434.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\overline K^0 n$ differential cross section. Data are from Prakhov 2009 \\cite{CrystalBall2005}, Conforto 1976 \\cite{Conforto1976}, and Griselin 1975 \\cite{Griselin1975}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_750.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_1153.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_2_1465.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ differential cross section. Data are from Prakhov 2009 \\cite{CrystalBall2005}, Armenteros 1968 \\cite{Armenteros1968}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_935.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_1245.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_31_1462.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^+\\Sigma^-$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_560.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_659.pdf}} \n\\vspace{-20mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_32_714.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Sigma^0$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970} and Prakhov 2009 \\cite{CrystalBall2005}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_495.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_935.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_1245.pdf}} \n\\vspace{-25mm}\n\\vspace{3mm}\n\\scalebox{0.35}{\\includegraphics{.\/dSigma_33_1462.pdf}}\n\\vspace{-2mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\nFigures 8, 9, 10, and 11 show representative energy-dependent fit results for the polarization in reactions $K^- p \\rightarrow K^- p$, $K^- p \\rightarrow \\pi\\Lambda$, $K^- p \\rightarrow \\pi^0\\Sigma^0 $, and $K^- p \\rightarrow \\pi^- \\Sigma^+$, respectively. The polarizations are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson. Figure 8 shows the excellent agreement of our energy-dependent solution with the $K^-p\\rightarrow K^-p$ polarization at $P_{\\rm Lab}$ = 1383, 1483, 1584, 1684 MeV from Ref. \\cite{Daum1968}. For $K^-p\\rightarrow\\pi^0\\Lambda$ (Fig.\\ 9) our solution is in good agreement with the polarization data at $P_{\\rm Lab}$ = 514, 936, and 1165 MeV. Our solution also agrees well the Crystal Ball data at $P_{\\rm Lab}$ = 714 MeV at forward angles but there is a slight under representation of data at backward angles. Similarly, Fig.\\ 10 shows a very good agreement between our solution and the $K^-p\\rightarrow \\pi^0\\Sigma^0$ polarization data within the given uncertainties at $P_{\\rm Lab}$ = 514, 581, 687, and 750 MeV, all from Crystal Ball Collaboration. Finally, Fig.\\ 11 shows a comparison of our solution with the $K^-p\\rightarrow \\pi^-\\Sigma^+$ polarization data at  $P_{\\rm Lab}$ = 862, 936, 1001, and 1125 MeV. Except for small forward angles at $P_{\\rm Lab}$ = 1001 MeV, we have good agreement with the data.\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1383.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1483.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1584.pdf}} \n\\vspace{-25mm}\n\\vspace{3mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_11_1684.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^-p$ differential cross section. Data are from Daum 1968 \\cite{Daum1968}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_714.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_936.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_2_1165.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_514.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_581.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_687.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_32_750.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_862.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_936.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_1001.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/Polar_33_1125.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^-\\Sigma^+$ differential cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}, Armenteros 1968 \\cite{Armenteros1968}, Conforto 1976 \\cite{Conforto1976}, and Berthon 1970 \\cite{Berthon1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\nFigures 12, 13, 14, and 15 show representative energy-dependent fit results for the polarized cross section in reactions $K^- p \\rightarrow K^- p$, $K^- p \\rightarrow \\pi\\Lambda$, $K^- p \\rightarrow \\pi^0\\Sigma^0 $, and $K^- p \\rightarrow \\pi^- \\Sigma^+$, respectively. The polarized cross sections are shown as a function of $\\cos\\theta$, where $\\theta$ is the c.m.\\ scattering angle of the meson.\nWithin the uncertainties associated with the polarized cross section data our results are in good agreement with the data for  these reactions.\n\n\\begin{comment}\n\\begin{figure}[htpb]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_865.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1082.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1330.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_11_1732.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ polarized cross section. Data are from Albrow 1971 \\cite{Albrow1971}, Andersson 1970 \\cite{Andersson1970}, and Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[htpb]\n\\vspace{-15mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_865.pdf}}\n\\vspace{-1mm}\n\\vspace{-25mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1082.pdf}}\n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1330.pdf}} \n\\vspace{-25mm}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/PdSigma_11_1732.pdf}}\n\\vspace{-5mm}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow K^- p$ polarized cross section. Data are from Albrow 1971 \\cite{Albrow1971}, Andersson 1970 \\cite{Andersson1970}, and Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[htpb]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_514.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_554.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_637.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_2_719.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Lambda$ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_495.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_597.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_699.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_32_793.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^0\\Sigma^0 $ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}.}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\\begin{figure}[H]\n\\vspace{-10mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_495.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_597.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_699.pdf}\n\\vspace{-1mm}\n\\vspace{-20mm}\n\\includegraphics[width=0.51\\textwidth]{PdSigma_33_793.pdf}\n\\caption{(Color online) Representative results of our energy-dependent fit for the $K^- p \\rightarrow \\pi^- \\Sigma^+$ polarized cross section. Data are from Armenteros 1970 \\cite{Armenteros1970}}\n\\label{fig:dSigma_11_New}\n\\end{figure}\n\n\n Figure 16 shows our prediction for the total $K^-p\\rightarrow$ cross section.\n Figure 17 shows our prediction for the $K^-p\\rightarrow K^-p$ and $K^-p\\rightarrow \\overline K^0n$ integrated cross sections, Fig.\\ 18 shows our prediction for the $K^-p\\rightarrow \\pi^0\\Lambda$ integrated cross section, and Fig. 19 shows our prediction for the $K^-p\\rightarrow \\pi^+\\Sigma^-$, $K^-p\\rightarrow \\pi^0\\Sigma^0$, and $K^-p\\rightarrow\\pi^-\\Sigma^+$ integrated cross sections. Figures 17 and 18 do not show predictions for c.m.\\ energies below 1540 MeV because it was not possible to obtain single-energy amplitudes in this region where only integrated cross-section data are available.\n\n\\begin{figure}[H]\n\\begin{center}\n\\vspace{-1mm}\n\\scalebox{0.35}{\\includegraphics{.\/All1.pdf}}\n\\caption{(Color online) Total $K^-p$ cross section compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n \n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_11.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma11_12.pdf}}\n\\vspace{-10mm}\n\\caption{(Color online) Integrated cross sections for $K^- p \\rightarrow K^-p$ and $K^- p \\rightarrow \\overline K^0n$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.08mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma21_2.pdf}}\n\\caption{(Color online) Integrated cross section for $K^- p \\rightarrow \\pi^0\\Lambda$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}.}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\vspace{-9.5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_31.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma1_32.pdf}}\n\\vspace{-10mm}\n\\vspace{-5mm}\n\\scalebox{0.35}{\\includegraphics{.\/Sigma31_33.pdf}}\n\\vspace{-10mm}\n\\caption{(Color online) Integrated cross sections for $K^- p \\rightarrow \\pi^+\\Sigma^-$, $K^- p \\rightarrow \\pi^0\\Sigma^0$, and $K^- p \\rightarrow \\pi^-\\Sigma^+$ compared with the results of our energy-dependent fit. Data are from Baldini 1988 \\cite{Baldini1988}.}\n\\label{fig:dSigma_11_New}\n\\end{center}\n\\end{figure}\n\n\\clearpage\n\\section{Summary and Conclusions}\nWe have investigated $\\overline KN\\rightarrow \\overline KN$, $\\overline K N\\rightarrow \\pi\\Lambda$, and $\\overline KN\\rightarrow \\pi\\Sigma$ reactions through single-energy analyses constrained by a global unitary energy-dependent fit from threshold to a c.m.\\ energy of 2.1 GeV. We found partial waves up to G-waves necessary to describe the available data for the reactions. This work was motivated, in part, by the relatively recent measurements for $K^-p\\rightarrow \\overline K^0n$, $K^-p\\rightarrow \\pi^0\\Lambda$, $K^-p\\rightarrow\\pi^0\\Sigma^0$, and $K^-p\\rightarrow \\eta\\Lambda$ by the Crystal Ball Collaboration. We were successful in describing these data in addition to older data from constrained single-energy analyses. The partial-wave amplitudes thus extracted were used in our global multichannel fit. A discussion of the resonance parameters from this global fit, which is the most comprehensive multichannel fit to date for $\\overline K N$ scattering reactions, is presented in a separate publication \\cite{manoj2013}. \n\n\n\\acknowledgements{This work was supported by the U.S. Department of Energy Grant No. DE-FG02-01ER41194. \n       \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}