FBSDE with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity

TL;DR

This paper extends the theory of backward stochastic differential equations with time delays, providing new estimates, existence conditions, differentiability results, and path regularity properties in various function spaces.

Contribution

It introduces $L^p$-estimates, existence criteria, and differentiability analysis for delay BSDEs and FBSDEs, extending classical results to the delayed setting.

Findings

Established $L^p$-moment estimates for delay BSDEs

Provided conditions for existence and differentiability of solutions

Extended $L^2$-path regularity to delay FBSDEs

Abstract

We extend the work of Delong and Imkeller (2010a,b) concerning Backward stochastic differential equations with time delayed generators (delay BSDE). We give moment and a priori estimates in general $L^p$-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in $L^p$. We introduce decoupled systems of SDE and delay BSDE (delay FBSDE) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDE via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic $L^2$-path regularity to delay FBSDE.