Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

TL;DR

This paper explores optimal stopping problems for dynamic risk measures modeled by BSDEs with jumps, establishing existence, uniqueness, and characterization results, and extending to robust cases with model ambiguity.

Contribution

It provides new existence, uniqueness, and comparison theorems for reflected BSDEs with jumps and links these to optimal stopping problems under model uncertainty.

Findings

Characterized the value function as a solution to RBSDEs with jumps

Proved existence of optimal stopping times under semi-continuity conditions

Extended the framework to robust stopping problems with model ambiguity

Abstract

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.