Set-Valued Backward Stochastic Differential Equations

TL;DR

This paper develops an analytic framework for set-valued backward stochastic differential equations, utilizing Hukuhara differences to address the lack of inverse operations and extending integrals for set-valued martingales.

Contribution

It introduces a novel framework for set-valued BSDEs using Hukuhara differences and extends integrals to represent set-valued martingales, advancing the mathematical foundation for dynamic set-valued risk measures.

Findings

Established well-posedness of a class of set-valued BSDEs

Extended Aumann-Itô integrals for set-valued martingales

Connected set-valued BSDEs with dynamic risk measures

Abstract

In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann-It\^o integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with non-singleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.