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

In this article, 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.