Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs

We study a “new kind” of backward doubly stochastic differential equations, where the nonlinear noise term is given by Ito–Kunita’s stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolevʹs solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise.