Pathwise Holder convergence of the implicit Euler scheme for semi-linear   SPDEs with multiplicative noise
  In this article we prove pathwise Holder convergence with optimal rates of the implicit Euler scheme for semi-linear parabolic stochastic differential equations with multiplicative noise, set in a UMD Banach space X. We assume the non-linearities to satisfy appropriate (local) Lipschitz conditions. The convergence results are obtained by first proving corresponding results for the splitting scheme. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise. 
S.G. Cox and J.M.A.M. van Neerven
math.FA math.NA math.PR