BSDEs on finite and infinite horizon with time-delayed generators

TL;DR

This paper studies backward stochastic differential equations with time delays in their generators, establishing existence, uniqueness, and convergence results, and linking them to forward-backward systems for complex multi-dimensional cases.

Contribution

It introduces new existence and uniqueness results for delayed BSDEs on finite and infinite horizons, and connects these to forward-backward systems for advanced applications.

Findings

Solutions with delay converge to non-delayed solutions as delay vanishes

Existence and uniqueness are established for delayed BSDEs with possibly infinite horizon

The approach extends to quadratic delayed equations in multi-dimension

Abstract

We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without reflection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, and we exemplify a situation where this observation allows to derive results for quadratic delayed equations with non-bounded terminal conditions in multi-dimension.