Backward doubly stochastic differential equations with polynomial growth coefficients

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coeffi-cients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.