  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.
