A note on Multiplicative Poisson Equation: developments in the span-contraction approach

In this paper we study the problem of Multiplicative Poisson Equation (MPE) bounded solution existence in the generic discrete-time setting. Assuming mixing and boundedness of the risk-reward function, we investigate what conditions should be imposed on the underlying non-controlled probability kernel or the reward function in order for the MPE bounded solution to always exists. In particular, we consolidate span-norm framework based results and derive an explicit sharp bound that needs to be imposed on the cost function to guarantee the bounded solution existence under mixing. Also, we study the properties which the probability kernel must satisfy to ensure existence of bounded MPE for any generic risk-reward function and characterise process behaviour in the complement of the invariant measure support. Finally, we present numerous examples and stochastic-dominance based arguments that help to better understand the intricacies that emerge when the ergodic risk-neutral mean operator is replaced with ergodic risk-sensitive entropy

Continuous-time Markov decision processes under the risk-sensitive average cost criterion

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.Comment: 14 page

Markov Decision Processes with Risk-Sensitive Criteria: An Overview

The paper provides an overview of the theory and applications of risk-sensitive Markov decision processes. The term 'risk-sensitive' refers here to the use of the Optimized Certainty Equivalent as a means to measure expectation and risk. This comprises the well-known entropic risk measure and Conditional Value-at-Risk. We restrict our considerations to stationary problems with an infinite time horizon. Conditions are given under which optimal policies exist and solution procedures are explained. We present both the theory when the Optimized Certainty Equivalent is applied recursively as well as the case where it is applied to the cumulated reward. Discounted as well as non-discounted models are reviewe