Minimizing Spectral Risk Measures Applied to Markov Decision Processes

TL;DR

This paper develops a method to minimize spectral risk measures in Markov Decision Processes, providing theoretical foundations, solution algorithms, and applications to reinsurance, extending previous work on risk measures like Expected Shortfall.

Contribution

It introduces a novel approach to optimize spectral risk measures in MDPs, including existence proofs and an algorithm for infinite-dimensional problems, extending prior research.

Findings

Inner minimization as an ordinary MDP on an extended state space

Existence of solutions for the outer minimization problem

Algorithm for numerical approximation of the optimal risk measure

Abstract

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 B\"auerle 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.