FBSDEs with time delayed generators: LP-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6,7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). 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 SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L-2-path regularity to delay FBSDEs. (C) 2011 Elsevier B.V. All rights reserved.