Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

TL;DR

This paper proves the existence and uniqueness of solutions for a complex class of stochastic differential equations involving forward-backward coupling, doubly stochastic terms, and Poisson jumps, under certain conditions.

Contribution

It introduces a novel approach to establish existence and uniqueness for fully coupled forward-backward doubly stochastic differential equations with jumps using a continuation method.

Findings

Proved existence and uniqueness under monotonicity conditions.

Established strong solutions for the complex stochastic system.

Extended the theory to include Poisson jumps in the equations.

Abstract

The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense that initial and terminal values depend on solutions, and are not just fixed random variables. We establish under some monotonicity conditions, the existence and uniqueness of strong solutions of such equations by using a continuation method.