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

We extend some works of Delong and Imkeller concerning Backward stochastic differential equations with time delayed generators (delay BSDE). We provide sharper a priori estimates and show that the solution of a delay BSDE is in $L^p$. We introduce decoupled systems of SDE and delay BSDE (which we term delay FBSDE) and give sufficient conditions for the variational differentiability of their solutions. We connect these derivatives to the Malliavin derivatives of such delay FBSDE via the usual representation formulas which in turn give access to several path regularity results. In particular we prove an extension of the $L^2$-path regularity result for delay FBSDE