On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

TL;DR

This paper extends the monotone stability approach to BSDEs driven by Brownian motion and jumps, providing verifiable criteria for existence, uniqueness, and bounds of solutions, with applications in finance and control.

Contribution

It introduces a generalized stability framework for BSDEs with jumps, relaxing convexity and Lipschitz conditions, and offers practical criteria and examples for their analysis.

Findings

Established criteria for existence and uniqueness of solutions.

Provided bounds and comparison results for BSDEs with jumps.

Illustrated applications in finance and optimal control.

Abstract

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.