We study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.
In the classical static optimal reinsurance problem, the cost of capital for the insurer's risk exposure determined by a monetary risk measure is minimized over the class of reinsurance treaties represented by increasing Lipschitz retained loss functions. In this paper, we consider a dynamic extension of this reinsurance problem in discrete time which can be viewed as a risk-sensitive Markov Decision Process. The model allows for both insurance claims and premium income to be stochastic and operates with general risk measures and premium principles. We derive the Bellman equation and show the …
We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. We derive risk-averse dynamic programming equations and we show that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We illustrate the results on an optimal stopping problem and an organ transplant problem.
We develop an approach to time-consistent risk evaluation of continuous-time processes in Markov systems. Our analysis is based on dual representation of coherent risk measures, differentiability concepts for multivalued mappings, and a refined concept of time consistency. We prove that the risk measures are defined by a family of risk evaluation functionals (transition risk mappings), which depend on state, time, and the transition function. Their dual representations are risk multikernels of the Markov system. We introduce the concept of a semi-derivative of a risk multikernel and use it to …
In this paper, we consider risk-sensitive discounted control problem for continuous-time jump Markov processes taking values in general state space. The transition rates of underlying continuous-time jump Markov processes and the cost rates are allowed to be unbounded. Under certain Lyapunov condition, we establish the existence and uniqueness of the solution to the Hamilton-Jacobi-Bellman (HJB) equation. Also we prove the existence of optimal risk-sensitive control in the class of Markov control.
In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.
In this article we consider risk-sensitive control of semi-Markov processes with a discrete state space. We consider general utility functions and discounted cost in the optimization criteria. We consider random finite horizon and infinite horizon problems. Using a state augmentation technique we characterise the value functions and also prescribe optimal controls.
We consider risk-sensitive Markov decision processes (MDPs), where the MDP model is influenced by a parameter which takes values in a compact metric space. We identify sufficient conditions under which small perturbations in the model parameters lead to small changes in the optimal value function and optimal policy. We further establish the robustness of the risk-sensitive optimal policies to modeling errors. Implications of the results for data-driven decision-making, decision-making with preference uncertainty, and systems with changing noise distributions are discussed.
This paper studies continuous-time Markov decision processes under the risk-sensitive average cost criterion. The state space is a finite set, the action space is a Borel space, the cost and transition rates are bounded, and the risk-sensitivity coefficient can take arbitrary positive real numbers. Under the mild conditions, we develop a new approach to establish the existence of a solution to the risk-sensitive average cost optimality equation and obtain the existence of an optimal deterministic stationary policy.
We introduce the Lyapunov approach to optimal control problems of average risk-sensitive Markov control processes with general risk maps. Motivated by applications in particular to behavioral economics, we consider possibly non-convex risk maps, modeling behavior with mixed risk preference. We introduce classical objective functions to the risk-sensitive setting and we are in particular interested in optimizing the average risk in the infinite-time horizon for Markov Control Processes on general, possibly non-compact, state spaces allowing also unbounded cost. Existence and uniqueness of an op…
We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.
In this article we consider zero and non-zero sum risk-sensitive average criterion games for semi-Markov processes with a finite state space. For the zero-sum case, under suitable assumptions we show that the game has a value. We also establish the existence of a stationary saddle point equilibrium. For the non-zero sum case, under suitable assumptions we establish the existence of a stationary Nash equilibrium.
In this paper, we propose an optimization framework that combines surveillance schedules and sparse control to bound the risk of spreading processes such as epidemics and wildfires. Here, risk is considered the risk of an undetected outbreak, i.e. the product of the probability of an outbreak and the impact of that outbreak, and we can bound or minimize the risk by resource allocation and persistent monitoring schedules. The presented framework utilizes the properties of positive systems and convex optimization to provide scalable algorithms for both surveillance and intervention purposes. We …
We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterise the value function via HJB equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.
We consider a risk-sensitive continuous-time Markov decision process over a finite time duration. Under the conditions that can be satisfied by unbounded transition and cost rates, we show the existence of an optimal policy, and the existence and uniqueness of the solution to the optimality equation out of a class of possibly unbounded functions, to which the Feynman-Kac formula was also justified to hold.
We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded. Under a certain Lyapunov stability assumption on the dynamics, we establish the existence of a value and saddle point equilibrium. Further we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Finally, we present and analyze an illustrative example.
The Vehicle Routing Problem (VRP) has been widely studied throughout its history as a way of optimizing routes by minimizing distances, and the issue of risk in VRP has been received less attention, which is essential to increase transport safety, to reduce accident costs and to improve delivery reliability. In this way, this paper aims to support decision makers to plan routes for a road freight company considering both, i.e. logistics cost and safety. An analytical approach based on statistics was developed in which official government data of accidents occurrences and data from cargo insura…
The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minimizing setting is ruled out. The existence and concrete forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests are proven. The well known quantile hedging method as well as the classical Neyman-Pearson lemma are generaliz…
We consider a piecewise deterministic Markov decision process, where the expected exponential utility of total (nonnegative) cost is to be minimized. The cost rate, transition rate and post-jump distributions are under control. The state space is Borel, and the transition and cost rates are locally integrable along the drift. Under natural conditions, we establish the optimality equation, justify the value iteration algorithm, and show the existence of a deterministic stationary optimal policy. Applied to special cases, the obtained results already significantly improve some existing results i…
We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation.
We consider a microgrid with random load realization, stochastic renewable energy production, and an energy storage unit. The grid controller provides the total net load trajectory that the microgrid should present to the main grid and the microgrid must impose load shedding and renewable energy curtailment if necessary to meet that net load trajectory. The microgrid controller seeks to operate the local energy storage unit to minimize the risk of load shedding, and renewable energy curtailment over a finite time horizon. We formulate the problem of optimizing the operation of the storage unit…
In this paper we present an algorithm to compute risk averse policies in Markov Decision Processes (MDP) when the total cost criterion is used together with the average value at risk (AVaR) metric. Risk averse policies are needed when large deviations from the expected behavior may have detrimental effects, and conventional MDP algorithms usually ignore this aspect. We provide conditions for the structure of the underlying MDP ensuring that approximations for the exact problem can be derived and solved efficiently. Our findings are novel inasmuch as average value at risk has not previously bee…
In classical Markov Decision Processes (MDPs), action costs and transition probabilities are assumed to be known, although an accurate estimation of these parameters is often not possible in practice. This study addresses MDPs under cost and transition probability uncertainty and aims to provide a mathematical framework to obtain policies minimizing the risk of high long-term losses due to not knowing the true system parameters. To this end, we utilize the risk measure value-at-risk associated with the expected performance of an MDP model with respect to parameter uncertainty. We provide mixed…
We establish existence of controlled Markov chain of mean-field type with unbounded jump intensities by means of a fixed point argument using the Wasserstein distance. Using a Markov chain entropic backward SDE approach, we further suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with risk sensitive payoff functionals of mean-field type.
We study nonzero-sum stochastic games for continuous time Markov chains on a denumerable state space with risk sensitive discounted and ergodic cost criteria. For the discounted cost criterion we first show that the corresponding system of coupled HJB equations has an appropriate solution. Then under an additional additive structure on the transition rate matrix and payoff functions, we establish the existence of a Nash equilibrium in Markov strategies. For the ergodic cost criterion we assume a Lyapunov type stability assumption and a small cost condition. Under these assumptions we show that…
Mechanical components that are exposed to cyclic mechanical loading fail at loads that are well below the ultimate tensile strength. This process is known as fatigue. The failure time, that is the time when a first crack forms, is highly random. In this work we review some recent developments in the modelling of probabilistic failure times, understood as the time to the formation of a fatigue crack. We also discuss the how probabilistic models can be used in shape design with the design intent of optimizing the component's reliability. We give review a recent existence result for optimal shape…
In this paper, as a first step in examining the properties of a feasible portfolio subset that is characterized by budget and risk constraints, we assess the maximum and minimum of the investment concentration using replica analysis. To do this, we apply an analytical approach of statistical mechanics. We note that the optimization problem considered in this paper is the dual problem of the portfolio optimization problem discussed in the literature, and we verify that these optimal solutions are also dual. We also present numerical experiments, in which we use the method of steepest descent th…
We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.
We provide a new algorithm for solving Risk Sensitive Partially Observable Markov Decisions Processes, when the risk is modeled by a utility function, and both the state space and the space of observations is finite. This algorithm is based on an observation that the change of measure and the subsequent introduction of the information space that is used for exponential utility functions, can be actually extended for sums of exponentials if one introduces an extra vector parameter that tracks the "expected accumulated cost" that corresponds to each exponential. Since every increasing function c…
We introduce a general framework for measuring risk in the context of Markov control processes with risk maps on general Borel spaces that generalize known concepts of risk measures in mathematical finance, operations research and behavioral economics. Within the framework, applying weighted norm spaces to incorporate also unbounded costs, we study two types of infinite-horizon risk-sensitive criteria, discounted total risk and average risk, and solve the associated optimization problems by dynamic programming. For the discounted case, we propose a new discount scheme, which is different from …
This paper is devoted to solving a time-inconsistent risk-sensitive control problem with parameter $\e$ and its limit case ($\e\rightarrow0^+$) for countable-stated Markov decision processes (MDPs for short). Since the cost functional is time-inconsistent, it is impossible to find a global optimal strategy for both cases. Instead, for each case, we will prove the existence of time-inconstant equilibrium strategies which verify the so-called step-optimality. Moreover, we prove the convergence of $\e$-equilibriums and the corresponding value functions as $\e\rightarrow0^+$.
We consider zero-sum stochastic games for continuous time Markov decision processes with risk-sensitive average cost criterion. Here the transition and cost rates may be unbounded. We prove the existence of the value of the game and a saddle-point equilibrium in the class of all stationary strategies under a Lyapunov stability condition. This is accomplished by establishing the existence of a principal eigenpair for the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. This in turn is established by using the nonlinear version of Krein-Rutman theorem. We then obtain a characterization of th…
We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, while the state process and the controls are observed at time $t$, the actual cost that may depend on the unknown parameter is not known at time $t$. The controller optimizes total cost by using a family of special risk measures, that we call risk filters and that are appropriately defined to take into account the model uncertainty of the controlled system. These key features lead to non-standard and …
California Institute of Technology（加州理工学院）；NASA Jet Propulsion Laboratory（NASA喷气推进实验室）
We consider the problem of designing policies for Markov decision processes (MDPs) with dynamic coherent risk objectives and constraints. We begin by formulating the problem in a Lagrangian framework. Under the assumption that the risk objectives and constraints can be represented by a Markov risk transition mapping, we propose an optimization-based method to synthesize Markovian policies that lower-bound the constrained risk-averse problem. We demonstrate that the formulated optimization problems are in the form of difference convex programs (DCPs) and can be solved by the disciplined convex-…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Purdue University（普渡大学）
We develop a stochastic approximation-type algorithm to solve finite state/action, infinite-horizon, risk-aware Markov decision processes. Our algorithm has two loops. The inner loop computes the risk by solving a stochastic saddle-point problem. The outer loop performs $Q$-learning to compute an optimal risk-aware policy. Several widely investigated risk measures (e.g. conditional value-at-risk, optimized certainty equivalent, and absolute semi-deviation) are covered by our algorithm. Almost sure convergence and the convergence rate of the algorithm are established. For an error tolerance $ε>…
This paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem via a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). Through Lagrangian relaxation, we convert the chance-constrained (risk-constrained) SOC problem to a risk-minimizing SOC problem, the cost function of which possesses the time-additive Bellman structure. We show that the risk-minimizing control synthesis is equivalent to solving an HJB PDE whose boundary condition can be tuned appropriately to achieve a desired level of safety. Furthermore, it is shown that …
The infinite horizon risk-sensitive discounted-cost and ergodic-cost nonzero-sum stochastic games for controlled Markov chains with countably many states are analyzed. For the discounted-cost game, we prove the existence of Nash equilibrium strategies in the class of Markov strategies under fairly general conditions. Under an additional geometric ergodicity condition and a small cost criterion, the existence of Nash equilibrium strategies in the class of stationary Markov strategies is proved for the ergodic-cost game.
In this article, we study the ergodic risk-sensitive control problem for controlled regime-switching diffusions. Under a blanket stability hypothesis, we solve the associated nonlinear eigenvalue problem for weakly coupled systems and characterize the optimal stationary Markov controls via a suitable verification theorem. We also consider the near-monotone case and obtain the existence of principal eigenfunction and optimal stationary Markov controls.
We motivate and propose a new model for non-cooperative Markov game which considers the interactions of risk-aware players. This model characterizes the time-consistent dynamic "risk" from both stochastic state transitions (inherent to the game) and randomized mixed strategies (due to all other players). An appropriate risk-aware equilibrium concept is proposed and the existence of such equilibria is demonstrated in stationary strategies by an application of Kakutani's fixed point theorem. We further propose a simulation-based Q-learning type algorithm for risk-aware equilibrium computation. T…
This paper considers the problem of finding near-optimal Markovian randomized (MR) policies for finite-state-action, infinite-horizon, constrained risk-sensitive Markov decision processes (CRSMDPs). Constraints are in the form of standard expected discounted cost functions as well as expected risk-sensitive discounted cost functions over finite and infinite horizons. The main contribution is to show that the problem possesses a solution if it is feasible, and to provide two methods for finding an approximate solution in the form of an ultimately stationary (US) MR policy. The latter is achieve…
We consider statistical Markov Decision Processes where the decision maker is risk averse against model ambiguity. The latter is given by an unknown parameter which influences the transition law and the cost functions. Risk aversion is either measured by the entropic risk measure or by the Average Value at Risk. We show how to solve these kind of problems using a general minimax theorem. Under some continuity and compactness assumptions we prove the existence of an optimal (deterministic) policy and discuss its computation. We illustrate our results using an example from statistical decision t…
