A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations

TL;DR

This paper presents a representation theorem for dynamic capital allocation under Itô-Lévy models, focusing on BSDE-based risk measures with quadratic-exponential growth, and illustrates it with entropic and coherent risk measures.

Contribution

It introduces a new representation theorem for dynamic capital allocation derived from BSDEs with jumps, expanding understanding of risk measures in stochastic models.

Findings

Derived a representation for dynamic risk measures under Itô-Lévy models.

Established a capital allocation formula based on BSDE differentiability.

Applied results to entropic and coherent risk measures.

Abstract

In this paper, we provide a representation theorem for dynamic capital allocation under It{\^o}-L{\'e}vy model. We consider the representation of dynamic risk measures defined under Backward Stochastic Differential Equations (BSDE) with generators that grow quadratic-exponentially in the control variables. Dynamic capital allocation is derived from the differentiability of BSDEs with jumps. The results are illustrated by deriving a capital allocation representation for dynamic entropic risk measure and static coherent risk measure.