Anticipated backward stochastic differential equations

TL;DR

This paper introduces anticipated backward stochastic differential equations (anticipated BSDEs), which incorporate future solution values into their generators, and establishes their fundamental properties including existence, uniqueness, and duality with delay equations.

Contribution

The paper defines anticipated BSDEs, proves their well-posedness, and explores their relationship with stochastic delay differential equations, advancing the theory of stochastic differential equations.

Findings

Anticipated BSDEs have unique solutions.

A comparison theorem for anticipated BSDEs is established.

Duality between anticipated BSDEs and stochastic delay equations is demonstrated.

Abstract

In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.