Generalized Reflected BSDEs with RCLL Random Obstacles in a General Filtration

TL;DR

This work studies generalized reflected backward stochastic differential equations (GRBSDEs) in a general filtration that supports a Brownian motion and an independent integer-valued random measure, with data satisfying $\mathbb{L}^2$-integrability and monotonicity in $y$. The authors develop existence and uniqueness results for GRBSDEs with a single lower RCLL barrier using a penalization scheme, the Snell envelope, and optimal stopping techniques, and establish a Skorokhod-type minimality through a decomposition $K=K^c+K^d$. They extend the framework to GRBSDEs with an upper barrier, prove a comparison principle (including when the generator depends on the jump term), and derive a connection to associated optimal stopping problems, providing a robust tool for stochastic control with jumps in general filtrations. The results pave the way for analyzing doubly reflected GRBSDEs via penalization schemes and have potential applications in stochastic PDE representations with nonlinear boundary conditions under general information flows.

Abstract

This paper addresses the existence and uniqueness of solutions to Reflected Generalized Backward Stochastic Differential Equations (GRBSDEs) within a general filtration that supports a Brownian motion and an independent integer-valued random measure. Our study focuses on cases where the given data satisfy appropriate $\mathbb{L}^2$-integrability conditions and the coefficients satisfy a monotonicity assumption. Additionally, we establish a connection between the solution and an optimal control problem over the set of stopping times.

Theorems & Definitions (15)

• Lemma 1: Lemma III.4.24 in jacodshiryaev
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 6
• Theorem 7
• Lemma 8
• Lemma 9
• Lemma 10