# BSDEs and SDEs with time-advanced and -delayed coefficients

## TL;DR

This paper studies a new class of backward and forward stochastic differential equations with coefficients depending on present, past, and future states, establishing existence, uniqueness, and duality under small delay or advance conditions.

## Contribution

It introduces and analyzes BSDEs and SDEs with time-advanced and -delayed coefficients, including duality results, which extend traditional stochastic differential equation theory.

## Key findings

- Existence and uniqueness of BSDEs with time-advanced/delayed coefficients.
- Existence and uniqueness of SDEs with time-advanced/delayed coefficients.
- Duality between these BSDEs and SDEs.

## Abstract

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.