Set-Valued Dynamic Risk Measures for Processes and Vectors

TL;DR

This paper explores the relationship between set-valued risk measures for processes and vectors, establishing their equivalence and providing new dual representations and consistency conditions within a unified framework.

Contribution

It introduces an augmented framework for set-valued risk measures for processes, proving their equivalence to vector measures and deriving new dual representations and time consistency results.

Findings

Established equivalence between risk measures for processes and vectors.

Derived a new dual representation for set-valued risk measures.

Proposed an augmented definition for multiportfolio time consistency.

Abstract

The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast with scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilize this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multiportfolio time consistency between set-valued risk measures for processes and vectors is provided; to accomplish this, an augmented definition for multiportfolio time consistency of set-valued risk measures for processes is proposed.