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In this paper we address an open question formulated in [16] . That is, we extend the Itô–Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker–Planck equation associated to X , to random mappings f . To this end we provide new results on a class of adapted and non-adapted Fokker–Planck SPDEs and BSPDEs. [ABSTRACT FROM AUTHOR]

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]