A Boolean valued analysis approach to conditional risk

TL;DR

This paper introduces a Boolean valued analysis approach to relate classical convex risk measures to conditional risk measures, enabling transfer of properties and duality results between the two frameworks.

Contribution

It develops a transfer principle using Boolean valued analysis that interprets conditional risk measures as classical convex risk measures within a set-theoretic model, facilitating new dual representation theorems.

Findings

Established a general robust representation theorem for conditional risk measures

Provided a method to interpret duality theorems of convex risk measures as those of conditional risk measures

Demonstrated the applicability of the transfer principle to various cases

Abstract

By means of the techniques of Boolean valued analysis, we provide a transfer principle between duality theory of classical convex risk measures and duality theory of conditional risk measures. Namely, a conditional risk measure can be interpreted as a classical convex risk measure inside of a suitable set-theoretic model. As a consequence, many properties of a conditional risk measure can be interpreted as basic properties of convex risk measures. This amounts to a method to interpret a theorem of dual representation of convex risk measures as a new theorem of dual representation of conditional risk measures. As an instance of application, we establish a general robust representation theorem for conditional risk measures and study different particular cases of it.