$\mathbb{L}^p$-solutions $(1 <p< 2)$ for reflected BSDEs with general jumps and stochastic monotone generators

TL;DR

This work establishes the existence and uniqueness of $\mathbb{L}^p$-solutions for one-dimensional reflected BSDEs with jumps (RBSDEJs) in a Brownian-plus-Poisson framework for $p\in(1,2)$. It handles a general RCLL obstacle and a driver that is stochastically monotone in the state variable and stochastically Lipschitz in $(z,u)$, under suitable linear-growth and integrability conditions. The authors develop a two-pronged approach: first, a penalization method for the case when the driver does not depend on $(z,u)$ to obtain a priori estimates and existence, and then a fixed-point argument to address the full problem where $f$ depends on $(z,u)$. The results advance the theory of $\mathbb{L}^p$-solutions for RBSDEJs with monotone generators and general obstacles, with potential applications to stochastic control, finance, and nonlinear PDEs via probabilistic representations.

Abstract

We consider a one-reflected backward stochastic differential equation with a general RCLL barrier in a filtration that supports a Brownian motion and an independent Poisson random measure. We establish the existence and uniqueness of a solution in $\mathbb{L}^p$ for $p \in (1,2)$. The result is obtained by means of the penalization method, under the assumption that the coefficient is stochastically monotone with respect to the state variable $y$, stochastically Lipschitz with respect to the control variables $(z,u)$, and satisfies suitable linear growth and $p$-integrability conditions.

Theorems & Definitions (14)

• Definition 1
• Remark 2
• Remark 3
• Lemma 4
• Corollary 5
• Lemma 6
• Remark 7
• Lemma 8
• Proposition 9
• Proposition 10