Infinite Anticipation Backward Stochastic Differential Equations

TL;DR

This paper introduces a novel class of backward stochastic differential equations with infinite anticipation, establishing their unique solvability and applying them to solve stochastic control problems via duality.

Contribution

It presents the first study of BSDEs with infinite anticipation, proving existence, uniqueness, and comparison results, and links them to stochastic control with infinite delay.

Findings

Proved existence and uniqueness of solutions for infinite anticipation BSDEs.

Established a comparison theorem for these BSDEs.

Applied the theory to solve stochastic control problems with infinite delay.

Abstract

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs) with infinite anticipation, where the generator depends on the entire future values of the solution in infinite horizon. We show that the new BSDEs has a unique solution and admits a comparison result. In the end, we solve a stochastic control problem via a duality between BSDEs with infinite anticipation and stochastic differential equations (SDEs) with infinite delay.