THE OBSTACLE PROBLEM FOR QUASILINEAR STOCHASTIC PDES WITH NON-HOMOGENEOUS OPERATOR

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of ItO's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.