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Abstract: We extend the work of Delong and Imkeller (2010)  concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general -spaces and provide sufficient conditions for the solution of a delay BSDE to exist in . 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 -path regularity to delay FBSDEs. [Copyright &y& Elsevier]

In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces D1,p. [ABSTRACT FROM AUTHOR]