A Supermartingale Relation for Multivariate Risk Measures

TL;DR

This paper establishes a fundamental link between multivariate risk measures' time consistency and supermartingale properties, providing dual characterizations and examples that advance the understanding of dynamic risk assessment.

Contribution

It proves the equivalence between multiportfolio time consistency and a supermartingale property for multivariate risk measures, and characterizes the dual variables involved.

Findings

Supermartingale property characterizes time consistency.

Dual variables are identified as worst-case scenarios.

Examples of risk measures satisfying the property are provided.

Abstract

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.