We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model, we introduce a set of conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, which require compactness/continuity conditions on the MDP, our result does not and can be applied to problems with discontinuous dynamics and one-stage costs. The key idea here is to replace the compactness/continuity conditions used in the prior work by what we call majorization type conditions. In particular, among others, we require that for each state, on selected subsets of actions at that state, the state transition stochastic kernel is majorized by finite measures, and we use this majorization property together with Egoroff's theorem to prove the ACOI.
We also consider the minimum pair approach for average-cost MDPs and apply the majorization idea. For the case of a discrete action space and strictly unbounded costs, we prove the existence of a minimum pair that consists of a stationary policy and an invariant probability measure induced by the policy. This result is derived by combining Lusin's theorem with another majorization condition we introduce, and it can be applied to a class of countable action space MDPs in which, with respect to the state variable, the dynamics and one-stage costs are discontinuous.
We consider discrete-time Markov Decision Processes with Borel state and action spaces and universally measurable policies. For several long-run average cost criteria, we establish the following optimality results: the optimal average cost functions are lower semianalytic, there exist universally measurable semi-Markov or history-dependent $ε$-optimal policies, and similar results hold for the minimum average costs achievable by Markov or stationary policies. We then analyze the structure of the optimal average cost functions, proving sufficient conditions for them to be constant almost everyw…
We study discrete-time discounted constrained Markov decision processes (CMDPs) on Borel spaces with unbounded reward functions. In our approach the transition probability functions are weakly or set-wise continuous. The reward functions are upper semicontinuous in state-action pairs or semicontinuous in actions. Our aim is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of stationary randomized poli…
We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model with a Lyapunov-type stability character, we introduce a set of new conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, our result does not require any compactness and continuity conditions on the MDPs. Instead, the main idea is to use the almost-uniform-convergence property of a pointwise converge…
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.
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.
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.
The two expected average costs used in the theory of semi-Markov control processes with a Borel state space are considered. Under some stochastic stability conditions, we prove that the two criteria are equivalent in the sense that they lead to the same optimality equation.
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…
On Linear Programming for Constrained and Unconstrained Average-Cost Markov Decision Processes with Countable Action Spaces and Strictly Unbounded Costs
This article considers the average optimality for a continuous-time Markov decision process with Borel state and action spaces and an arbitrarily unbounded nonnegative cost rate. The existence of a deterministic stationary optimal policy is proved under a different and general set of conditions as compared to the previous literature; the controlled process can be explosive, the transition rates can be arbitrarily unbounded and are weakly continuous, the multifunction defining the admissible action spaces can be neither compact-valued nor upper semi-continuous, and the cost rate is not necessar…
This paper deals with unconstrained discounted continuous-time Markov decision processes in Borel state and action spaces. Under some conditions imposed on the primitives, allowing unbounded transition rates and unbounded (from both above and below) cost rates, we show the regularity of the controlled process, which ensures the underlying models to be well defined. Then we develop the dynamic programming approach by showing that the Bellman equation is satisfied (by the optimal value). Finally, under some compactness-continuity conditions, we obtain the existence of a deterministic stationary …
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…
The present paper considers the constrained optimal control problem with total undiscounted criteria for a continuous-time Markov decision process (CTMDP) in Borel state and action spaces. Under the standard compactness and continuity conditions, we show the existence of an optimal stationary policy out of the class of general nonstationary ones. In the process, we justify the reduction of the CTMDP model to a discrete-time Markov decision process (DTMDP) model based on the studies of the undiscounted occupancy and occupation measures. We allow that the controlled process is not necessarily ab…
In this paper, we consider a continuous-time Markov decision process (CTMDP) in Borel spaces, where the certainty equivalent with respect to the exponential utility of the total undiscounted cost is to be minimized. The cost rate is nonnegative. We establish the optimality equation. Under the compactness-continuity condition, we show the existence of a deterministic stationary optimal policy. We reduce the risk-sensitive CTMDP problem to an equivalent risk-sensitive discrete-time Markov decision process, which is with the same state and action spaces as the original CTMDP. In particular, the v…
This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincaré inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.
This paper presents sufficient conditions for the existence of stationary optimal policies for average-cost Markov Decision Processes with Borel state and action sets and with weakly continuous transition probabilities. The one-step cost functions may be unbounded, and action sets may be noncompact. The main contributions of this paper are: (i) general sufficient conditions for the existence of stationary discount-optimal and average-cost optimal policies and descriptions of properties of value functions and sets of optimal actions, (ii) a sufficient condition for the average-cost optimality o…
This paper, based on the compactness-continuity and finite value conditions, establishes the sufficiency of the class of stationary policies out of the general class of history-dependent ones for a constrained continuous-time Markov decision process in Borel state and action spaces with total nonnegative cost criteria. The controlled process is not necessarily absorbing, the discount factor can be identically equal to zero, and the transition rates can also be equal to zero, which account for the major technical difficulties. Models with these features seemingly had not been handled in the pre…
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…
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.
We study a finite-horizon two-person zero-sum risk-sensitive stochastic game for continuous-time Markov chains and Borel state and action spaces, in which payoff rates, transition rates and terminal reward functions are allowed to be unbounded from below and from above and the policies can be history-dependent. Under suitable conditions, we establish the existence of a solution to the corresponding Shapley equation (SE) by an approximation technique. Then, by the SE and the extension of the Dynkin's formula, we prove the existence of a Nash equilibrium and verify that the value of the stochast…
This paper describes sufficient conditions for the existence of optimal policies for Partially Observable Markov Decision Processes (POMDPs) with Borel state, observation, and action sets and with the expected total costs. Action sets may not be compact and one-step cost functions may be unbounded. The introduced conditions are also sufficient for the validity of optimality equations, semi-continuity of value functions, and convergence of value iterations to optimal values. Since POMDPs can be reduced to Completely Observable Markov Decision Processes (COMDPs), whose states are posterior state…
In this work, we study discrete-time Markov decision processes (MDPs) under constraints with Borel state and action spaces and where all the performance functions have the same form of the expected total reward (ETR) criterion over the infinite time horizon. One of our objective is to propose a convex programming formulation for this type of MDPs. It will be shown that the values of the constrained control problem and the associated convex program coincide and that if there exists an optimal solution to the convex program then there exists a stationary randomized policy which is optimal for th…
Sun Yat-Sen University（中山大学）；Stanford University（斯坦福大学）
CVaR (Conditional Value at Risk) is a risk metric widely used in finance. However, dynamically optimizing CVaR is difficult since it is not a standard Markov decision process (MDP) and the principle of dynamic programming fails. In this paper, we study the infinite-horizon discrete-time MDP with a long-run CVaR criterion, from the view of sensitivity-based optimization. By introducing a pseudo CVaR metric, we derive a CVaR difference formula which quantifies the difference of long-run CVaR under any two policies. The optimality of deterministic policies is derived. We obtain a so-called Bellma…
We aim at the construction of a Hidden Markov Model (HMM) of assigned complexity (number of states of the underlying Markov chain) which best approximates, in Kullback-Leibler divergence rate, a given stationary process. We establish, under mild conditions, the existence of the divergence rate between a stationary process and an HMM. Since in general there is no analytic expression available for this divergence rate, we approximate it with a properly defined, and easily computable, divergence between Hankel matrices, which we use as our approximation criterion. We propose a three-step algorith…
We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a family of ordinary differential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A sufficient criterion for this stronger notion is also provided. The result…
A multiplicative relative value iteration algorithm for solving the dynamic programming equation for the risk-sensitive control problem is studied for discrete time controlled Markov chains with a compact Polish state space, and controlled diffusions in on the whole Euclidean space. The main result is a proof of convergence to the desired limit in each case.
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 …
