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We consider a quasilinear parabolic stochastic partial differential equationdriven by a multiplicative noise and study regularity properties of its weaksolution satisfying classical a priori estimates. In particular, we determineconditions on coefficients and initial data under which the weak solution isH\"older continuous in time and possesses spatial regularity that is onlylimited by the regularity of the given data. Our proof is based on an efficientmethod of increasing regularity: the solution is rewritten as the sum of twoprocesses, one solves a linear parabolic SPDE with the same noise term as theoriginal model problem whereas the other solves a linear parabolic PDE withrandom coefficients. This way, the required regularity can be achieved byrepeatedly making use of known techniques for stochastic convolutions anddeterministic PDEs.

A regularity result for quasilinear stochastic partial differential equations of parabolic type