Representation of BSDE-Based Dynamic Risk Measures and Dynamic Capital Allocations

Abstract

In this short paper we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations. We derive this representation from a classical differentiability result for backward stochastic differential equations and the full allocation property of the Aumann-Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our result are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gateaux-differentiable.