Coupled forward-backward stochastic differential equations with jumps in random environments

TL;DR

This paper establishes existence and uniqueness results for coupled forward-backward stochastic differential equations with jumps in a random environment, relevant to mean-field games and processes with common noise.

Contribution

It introduces a framework for FBSDEs with jumps in a random environment, extending models to include environment-dependent jumps and connections to Cox and Hawkes processes.

Findings

Proved existence and uniqueness of solutions.

Connected the model to Cox and Hawkes processes.

Related to regime-switching Conditional McKean-Vlasov equations.

Abstract

In this paper we obtain results for the existence and uniqueness of solutions to coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with jumps defined on a random environment. This environment corresponds to a measured-valued process, similar to the one found in Conditional McKean-Vlasov Differential Equations and Mean-Field Games with Common Noise. The jump term in the FBSDE is dependent on the environment through a stochastic intensity process. We provide examples which relate our model with FBSDEs driven by Cox and Hawkes processes, as well as regime-switching Conditional McKean-Vlasov differential equations.