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

Abstract

The paper addresses a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. These equations are allowed to live in Euclidean spaces of different dimensions, and the system is Markovian in the sense that the terminal value of the backward equation depends on the terminal value of the solution of the forward one. Under some monotonicity conditions we establish the existence and uniqueness of strong solutions of such equations by using a continuation method.

Stochastic time-optimal control and sensitivity studies for additional food provided prey-predator systems involving Holling type-IV functional response

“…In this section, we now prove the existence of optimal controls by proving the existence of the solutions for the FBSDEJ [( 5), ( 8), ( 12)] which establishes the existence of (x * , y * , p * 1 , p * 2 ) for all simulation purposes. Using the theorem in Al-Hussein and Gherbal [34], we now prove the existence of the optimal controls (15) in the following theorem. Theorem 2.…”

“…Following the the existence and uniqueness theorem of FBDSDEJ in Al-Hussein and Gherbal [34], we are only left to prove the monotonicity and Lipschitz continuity of the drift and diffusion terms of the adjoint system of Equation 8.…”