Martingale Characterizations of Risk-Averse Stochastic Optimization Problems

TL;DR

This paper explores how nested risk measures can be characterized using martingale properties to improve the formulation and analysis of risk-aware stochastic optimization problems, extending classical dynamic programming equations.

Contribution

It introduces martingale characterizations of nested risk measures and demonstrates their continuity properties within risk-aware stochastic optimization frameworks.

Findings

Martingale properties of nested risk measures are established.

Continuity of risk-aware optimization problems with respect to nested distance.

Extension of Hamilton-Jacobi-Bellman equations to include risk measures.

Abstract

This paper addresses risk awareness of stochastic optimization problems. Nested risk measures appear naturally in this context, as they allow beneficial reformulations for algorithmic treatments. The reformulations presented extend usual Hamilton-Jacobi-Bellman equations in dynamic optimization by involving risk awareness in the problem formulation. Nested risk measures are built on risk measures, which originate by conditioning on the history of a stochastic process. We derive martingale properties of these risk measures and use them to prove continuity. It is demonstrated that stochastic optimization problems, which incorporate risk awareness via nesting risk measures, are continuous with respect to the natural distance governing these optimization problems, the nested distance.