Robust optimal problem for dynamic risk measures governed by BSDEs with jumps and delayed generator

TL;DR

This paper investigates an optimal stopping problem for dynamic risk measures modeled by BSDEs with jumps and delays, establishing connections to reflected BSDEs and addressing ambiguity through a game-theoretic approach.

Contribution

It introduces a novel framework linking dynamic risk measures with reflected BSDEs involving jumps and delays, and solves the associated optimal stopping and game problems.

Findings

Established existence and uniqueness of solutions for the reflected BSDEs with jumps and delays.

Connected the value function of the stopping problem to reflected BSDEs with jumps and delays.

Addressed ambiguity in risk measures through a mixed/optimal stopping game approach.

Abstract

The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and delayed generator. Firstly, we connect the value function of this problem to reflected BSDEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case.