Process-Based Risk Measures and Risk-Averse Control of Discrete-Time Systems

TL;DR

This paper introduces process-based dynamic risk measures for discrete-time stochastic systems, establishing their structure, properties, and applications to controlled Markov processes with dynamic programming solutions.

Contribution

It develops a new class of risk measures that depend on process history, introduces stochastic time consistency, and provides a representation and dynamic programming framework.

Findings

Process-based risk measures can be represented by static law-invariant risk measures.

A new concept of stochastic time consistency is introduced.

Dynamic programming equations are derived for controlled Markov processes.

Abstract

For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main features are that they measure risk of processes that are functions of the history of a base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based risk measures enjoying this property. We show that they can be equivalently represented by a collection of static law-invariant risk measures on the space of functions of the state of the base process. We apply this result to controlled Markov processes and we derive dynamic programming equations.