$L^{p}$-solutions of backward stochastic differential equations with time-delayed generators

TL;DR

This paper investigates backward stochastic differential equations with time-delayed generators, establishing existence, uniqueness, and new estimates for solutions under $L^{p}$-integrability, simplifying previous proofs and providing explicit conditions.

Contribution

It introduces new estimates and simplified proofs for the existence and uniqueness of solutions to BSDEs with delayed generators under $L^{p}$-conditions.

Findings

Established new estimation techniques for BSDEs with delays

Proved existence and uniqueness under explicit conditions

Simplified previous proof methods

Abstract

This article is devoted to study the class of backward stochastic differential equation with delayed generator. We suppose the terminal value and the generator to be $L^{p}$-integrable with $p>1$. We derive a new type of estimation related to this BSDE. Next, we establish the existence and uniqueness result in two ways. First, an approximation technics used by Briand et al. (Stochastic Process. Appl. 108 (2003) 109-129) and hence the well-know Picard iterative procedure. Using Picard iterative procedure, we revisit the result of Dos Reis et al. (Stochastic Process. Appl. 121 (9) (2011) 2114-2150), simplifying the proof and give an explicit existence and uniqueness condition related to the Lipschitz constant $K$ and the terminal time $T$.