Backward doubly stochastic differential equations and systems of quasilinear SPDEs

A new kind of backward stochastic differential equations (in short BSDE), where the solution is a pair of processes adapted to the past of the driving Brownian motion, has been introduced by the authors in [63. It was then shown in a series of papers by the second and both authors (see [8, 7, 9, 103), that this kind of backward SDEs gives a probabilistic representation for the solution of a large class of systems of quasi-linear parabolic PDEs, which generalizes the classical Feynman-Kac formula for linear parabolic PDEs. On the other hand, the classical Feynman-Kac formula has been generalized by the first author in [-4, 5] to provide a probabilistic representation for solutions of linear parabolic stochastic partial differential equations; see also Krylov and Rozovskii [-1], Rozovskii [-11] and Ocone and Pardoux [-3] for further extensions. The aim of this paper is to combine the two above types of results, and relate a new class of backward stochastic differential equations, which we call "doubly stochastic" for reasons which will become clear below, to a class of systems of quasilinear parabolic SPDEs. Hence we shall give a probabilistic representation of solutions of such systems of quasilinear SPDEs, and use it to prove an existence and uniqueness result of such SPDEs.