$\mathbb{L}^p$ $(p>1)$-solutions for BSDEs with jumps and stochastic monotone generator

TL;DR

This paper proves existence and uniqueness of $ ext{L}^p$-solutions for multidimensional BSDEs with jumps, under stochastic monotonicity and integrability conditions, extending the theory to cases where $p$ is greater than 1.

Contribution

It establishes the first comprehensive existence and uniqueness results for $ ext{L}^p$-solutions of BSDEs with jumps under stochastic monotonicity for both $p eq 2$ and $p eq 1$.

Findings

Proved existence and uniqueness of $ ext{L}^p$-solutions for $p eq 2$.

Extended BSDE theory to include stochastic monotone generators with jumps.

Provided integrability conditions ensuring well-posedness.

Abstract

We study multidimensional discontinuous backward stochastic differential equations in a filtration that supports both a Brownian motion and an independent integer-valued random measure. Under suitable $\mathbb{L}^p$-integrability conditions on the data, we establish the existence and uniqueness of $\mathbb{L}^p$-solutions for both cases: $p \geq 2$ and $p \in (1,2)$. The generator is assumed to be stochastically monotone in the state variable $y$, stochastically Lipschitz in the control variables $(z, u)$, and to satisfy a stochastic linear growth condition, along with an appropriate $\mathbb{L}^p$-integrability requirement.