Anticipated backward doubly stochastic differential equations

TL;DR

This paper introduces anticipated backward doubly stochastic differential equations (anticipated BDSDEs), establishing their existence, uniqueness, and comparison theorems, and explores their duality with delayed doubly stochastic differential equations.

Contribution

It presents the first study of anticipated BDSDEs, including foundational theorems and a duality relationship with delayed DSDEs.

Findings

Proved existence and uniqueness of solutions for anticipated BDSDEs.

Established a comparison theorem for solutions.

Demonstrated duality between anticipated BDSDEs and delayed DSDEs.

Abstract

In this paper, we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations (anticipated BDSDEs). The coefficients of these BDSDEs depend on the future value of the solution $(Y, Z)$. We obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations. Besides, as an application, we also establish a duality between the anticipated BDSDEs and the delayed doubly stochastic differential equations (delayed DSDEs).