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The cyber risk insurance market is at a nascent stage of its development, even as the magnitude of cyber losses is significant and the rate of cyber loss events is increasing. Existing cyber risk insurance products as well as academic studies have been focusing on classifying cyber loss events and developing models of these events, but little attention has been paid to proposing insurance risk transfer strategies that incentivise mitigation of cyber loss through adjusting the premium of the risk transfer product. To address this important gap, we develop a Bonus-Malus model for cyber risk insurance. Specifically, we propose a mathematical model of cyber risk insurance and cybersecurity provisioning supported with an efficient numerical algorithm based on dynamic programming. Through a numerical experiment, we demonstrate how a properly designed cyber risk insurance contract with a Bonus-Malus system can resolve the issue of moral hazard and benefit the insurer. [ABSTRACT FROM AUTHOR]

In this paper we explore optimal liquidation in a market populated by a number of heterogeneous market makers that have limited inventory-carrying and risk-bearing capacity. We derive a reduced form model for the dynamics of their aggregated inventory considering a proper scaling limit. The resulting price impact profile is shown to depend on the characteristics and relative importance of their inventories. The model is flexible enough to reproduce the empirically documented power law behavior of the price impact function. For any choice of the market makers characteristics, optimal execution within this modeling approach can be recast as a linear-quadratic stochastic control problem. The value function and the associated optimal trading rate can be obtained semi-explicitly subject to solving a differential matrix Riccati equation. Numerical simulations are conducted to illustrate the performance of the resulting optimal liquidation strategy in relation to standard benchmarks. Remarkably, they show that the increase in performance is determined by a substantial reduction of higher order moment risk. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic optimal control of a tri-stable energy harvester (TEH) under a standard rectifier circuit driven by correlated colored noises is considered. From the view of physical intuition, the control force is directly separated into conservative component and dissipative one. Then, the stationary probability density function (SPDF) and the DC power of the controlled electromechanical coupled TEH can be derived by using the stochastic averaging based on energy-dependent frequency. The weighted combination of mean DC power and rectification efficiency is regarded as a performance index to transform the stochastic optimal control problem into an extremum problem of a multivariable function. The stochastic direct optimal control strategy developed in this paper avoids the trouble of dealing with complex differential equations. In contrast with the uncontrolled case, the TEH under direct optimal control achieves significant optimization of harvesting performance. The cross-correlation between the additive and multiplicative colored noises can break the symmetry of SPDF to induce random transition. The power conversion efficiency of the harvesting system can be optimized by choosing the appropriate noise intensities. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

We propose a refined dynamic programming model based on a hidden Markov chain formulation and a nonlinear filtering technique to calculate the optimal debt ratio for public and private sectors for different scenarios. We then conduct the empirical analysis of the U.S. markets in real estate and equities during 1991.Q1 and 2020.Q1, comparing them with the theoretical results. It indicates that U.S. households and governments spent more than they can afford. While households reduced their debt ratio during times of economic distress, the public sector hiked its debt ratio to stimulate the economy. The policy effect took a long time to accumulate, and the outcome was lower than expected to revitalize the economy in time. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an optimal dividend and reinsurance problem for an insurance company facing two types of risks: unstable income and potential loses. The arrival of all loses is characterized as a compound Poisson process. We assumes that every possible loss can be reinsured for a part of it. The reserve is a combination of a diffusion process and a controllable compound Poisson process. We investigate the optimal dividend and reinsurance strategy by analyzing the corresponding variational inequality on the value function. A significant difference from the existing literature is that the HJB equation in this variational inequality is a partial integro-differential equation with a functional optimization problem appearing in the integral operator. We not only prove the existence of a classical solution to the problem and the continuity, strict monotonicity, boundedness of the dividend free boundary, but also discuss the properties of the optimal reinsurance policy, including the continuity, monotonicity of the optimal part covered by reinsurance for each possible loss, and the smoothness of the reinsurance free boundary. [ABSTRACT FROM AUTHOR]

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators. [ABSTRACT FROM AUTHOR]

This paper studies a defined contribution (DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally share the difference between accumulated contributions and returned premiums. We suppose that the financial market consists of a risk-free asset, a stock, and a zero-coupon bond. The interest rate is driven by a stochastic affine interest rate model and the stock price follows the Heston's stochastic volatility model with stochastic interest rates. Different fund managers have different risk preferences, and the hyperbolic absolute risk aversion (HARA) utility function is a general one including a power utility, an exponential utility, and a logarithm utility as special cases. We are concerned with an optimal portfolio to maximize the expected utility of terminal wealth by choosing the HARA utility function in the analysis. By using the principle of dynamic programming and Legendre transform-dual theory, we obtain explicit solutions of optimal strategies. Some special cases are also derived in detail. Finally, a numerical simulation is provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems. [ABSTRACT FROM AUTHOR]

We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the Hölder exponent of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jumps. We illustrate this approach in the context of a display advertising auction problem. [ABSTRACT FROM AUTHOR]

This paper considers an optimal control problem for a time-invariant linear stochastic system with discrete time, scalar unbounded control, additive noise, and a probabilistic criterion for retaining its trajectories in a given neighborhood of zero. We use dynamic programming and two-sided Bellman function estimates to derive analytical expressions for the optimal control at two time steps and a suboptimal control on any control horizon. The effectiveness of these controls is illustrated on a numerical example. [ABSTRACT FROM AUTHOR]

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ( n ≥ 3 ). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided. [ABSTRACT FROM AUTHOR]

This paper investigates a continuous-time portfolio optimization problem with the following features: (i) a no-short selling constraint; (ii) a leverage constraint, that is, an upper limit for the sum of portfolio weights; and (iii) a performance criterion based on the lower mean square error between the investor's wealth and a predetermined target wealth level. Since the target level is defined by a deterministic function independent of market indices, it corresponds to the criterion of absolute return funds. The model is formulated using the stochastic control framework with explicit boundary conditions. The corresponding Hamilton–Jacobi–Bellman equation is solved numerically using the kernel-based collocation method. However, a straightforward implementation does not offer a stable and acceptable investment strategy; thus, some techniques to address this shortcoming are proposed. By applying the proposed methodology, two numerical results are obtained: one uses artificial data, and the other uses empirical data from Japanese organizations. There are two implications from the first result: how to stabilize the numerical solution, and a technique to circumvent the plummeting achievement rate close to the terminal time. The second result implies that leverage is inevitable to achieve the target level in the setting discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon [0, T] as T → ∞. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems. In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking the deterministic situations, it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem. [ABSTRACT FROM AUTHOR]

Method of synthesis of control of stochastic processes is considered. Compared to the known approaches, the proposed method provides significantly more efficient process control by optimizing according to criteria that are nonlinearly dependent on the density of the distribution of the process and not on its particular characteristics (expectation, variance, etc.). The procedure for its computational implementation has been developed. A numerical example of optimal control synthesis illustrating the effectiveness of the proposed method is given. The advantageous features of the developed approach provide the possibility of its wide practical application in modern and promising information and control systems. [ABSTRACT FROM AUTHOR]

In this paper we study optimal trading strategies in a financial market in which stock returns depend on a hidden Gaussian mean reverting drift process. Investors obtain information on that drift by observing stock returns. Moreover, expert opinions in the form of signals about the current state of the drift arriving at fixed and known dates are included in the analysis. Drift estimates are based on Kalman filter techniques. They are used to transform a power utility maximization problem under partial information into an optimization problem under full information where the state variable is the filter of the drift. The dynamic programming equation for this problem is studied and closed-form solutions for the value function and the optimal trading strategy of an investor are derived. They allow to quantify the monetary value of information delivered by the expert opinions. We illustrate our theoretical findings by results of extensive numerical experiments. [ABSTRACT FROM AUTHOR]

The paper models the maintenance of ecological networks in forest environments, built from bioreserves, patches and corridors, when these grids are subject to random processes such as extreme natural events. It also outlines a management plan to support the optimized results. After presenting the random graph-theoretic framework, we apply the stochastic optimal control to the graph dynamics. Our results show that the preservation of the network architecture cannot be achieved, under stochastic control, over the entire duration. It can only be accomplished, at the cost of sacrificing the links between the patches, by increasing the usage of the control devices. This would have a negative effect on the species migration by causing congestion among the channels left at their disposal. The optimal scenario, in which the shadow price is at its lowest and all connections are well-preserved, occurs at half of the course, be it the only optimal stopping moment found on the stochastic optimal trajectories. In such a scenario, the optimal forestry management policy has to integrate agility, integrated response, and quicker response time. [ABSTRACT FROM AUTHOR]

This work deals with an optimal benefit distribution and asset allocation problem for a defined contribution (DC) pension plan during its decumulation phase. With the phenomenon of longevity, the time horizon of pension management during this phase might be long, thus the influence of inflation is considered in the context. The inflation index is subjected to a Poisson jump and a Brownian uncertainty. Motivated by the work of Wang et al. (2018), it is assumed that the scheme provides cost of living adjustment, which is extended to a jump diffusion process in this work. The plan aims to reduce fluctuations of benefit and terminal wealth by investing the fund in a financial market consisting of a bank account, an inflation indexed bond and a stock. The dynamics of two risky assets are also given by jump diffusion processes. The closed form decisions are derived by using the dynamic programming approach. [ABSTRACT FROM AUTHOR]

The problem of minimizing or maximizing the time spent by a controlled diffusion process in a given interval is known as LQG homing. The optimal control, when it is possible to obtain an explicit solution to such a problem, is often expressed as special functions. Here, the inverse problem is considered: we determine, under certain assumptions, the processes for which the optimal control is a simple power function. Moreover, the problem is extended to the case of jump-diffusion processes. [ABSTRACT FROM AUTHOR]

This paper is to establish a sufficient maximum principle for one kind of stochastic optimal control problem with three types of delays: a discrete delay, a moving-average delay and a noisy memory. The main features of this research include the introduction of a unified adjoint equation and a simple method to get the adjoint process. One kind of optimal consumption problem and its special cases are studied as illustrative examples, for which the adjoint equations are solved with two different approaches and the optimal consumption strategies are obtained. [ABSTRACT FROM AUTHOR]

In this paper, inspired by various stochastic linear-quadratic (LQ, for short) problems, we develop the method of continuation to study nonlinear forward–backward stochastic differential equations (FBSDEs, for short) in a kind of domination-monotonicity frameworks. The coupling of FBSDEs is in a general form, i.e., it not only appears in integral terms and terminal terms, but also in initial terms. By virtue of introducing various matrices, matrix-valued random variables and matrix-valued stochastic processes, we present the domination-monotonicity framework carefully and rigorously. A unique solvability result and a pair of estimates for coupled FBSDEs are obtained (see Theorem 3.5 in the case of simple domination-monotonicity conditions and Theorem 5.2 in the case of multi-level self-similar domination-monotonicity structures). As applications of theoretical results, the related stochastic Hamiltonian systems of several LQ problems are discussed. [ABSTRACT FROM AUTHOR]

The evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with applications in IT and neurogeometry [see Citti et al. (SIAM J Imag Sci 9(1):212–237, 2016)]. Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we obtain a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr et al. (Commun Pure Appl Math 9(2):307–326, 2010). [ABSTRACT FROM AUTHOR]

Correlated with the trend of increasing degrees of freedom in robotic systems is a similar trend of rising interest in spatio-temporal systems described by partial differential equations (PDEs) among the robotics and control communities. These systems often exhibit dramatic under-actuation, high dimensionality, bifurcations, and multimodal instabilities. Their control represents many of the current-day challenges facing the robotics and automation communities. Not only are these systems challenging to control, but the design of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimization, or apply tools from linear systems theory under restrictive linearity assumptions in order to arrive at a control solution. This manuscript provides a novel sampling-based stochastic optimization framework based entirely in Hilbert spaces suitable for the general class of semi-linear SPDEs which describes many systems in robotics and applied physics. This framework is utilized for simultaneous policy optimization and actuator co-design optimization. The resulting algorithm is based on variational optimization, and performs joint episodic optimization of the feedback control law and the actuation design over episodes. We study first and second order systems, and in doing so, extend several results to the case of second order SPDEs. Finally, we demonstrate the efficacy of the proposed approach with several simulated experiments on a variety of SPDEs in robotics and applied physics including an infinite degree-of-freedom soft robotic manipulator. [ABSTRACT FROM AUTHOR]

This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray–Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, R δ -set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results. [ABSTRACT FROM AUTHOR]

A kind of linear-quadratic Stackelberg games with the multilevel hierarchy driven by both Brownian motion and Poisson processes is considered. The Stackelberg equilibrium is presented by linear forward-backward stochastic differential equations (FBSDEs) with Poisson processes (FBSDEPs) in a closed form. By the continuity method, the unique solvability of FBSDEPs with a multilevel self-similar domination-monotonicity structure is obtained. [ABSTRACT FROM AUTHOR]