Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations

TL;DR

This paper explores the extension of scalar backward stochastic differential equations to the multivariate, set-valued context using difference inclusions and equations in discrete time, aiming to inform continuous-time risk measure models.

Contribution

It introduces a discrete-time framework with difference inclusions and equations to analyze set-valued dynamic risk measures, bridging the gap to continuous-time models.

Findings

Discrete-time difference inclusions model set-valued risk dynamics.

Difference equations provide a computational approach for set-valued risk measures.

Insights gained may inform the development of continuous-time set-valued risk models.

Abstract

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.