Optimal Stopping of BSDEs with Constrained Jumps and Related Zero-Sum Games

TL;DR

This paper develops a non-linear Snell envelope for BSDEs with constrained jumps, establishing its existence and linking it to a zero-sum stochastic differential game involving impulse control and stopping strategies.

Contribution

It introduces a novel non-linear Snell envelope for constrained jump BSDEs and connects it to a new class of zero-sum stochastic differential games.

Findings

Existence of the non-linear Snell envelope proven.

Characterization of the game value via the Snell envelope.

New techniques in control randomization developed.

Abstract

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.