On the splitting-up method and stochastic partial differential equations

We consider two stochastic partial differential equations \[ duε(t)= (L_ruε(t)+fᵣ(t)) \,dVε t^r+(Mₖuε(t)+g_k(t))\, ∘ dY_t^k, *{-5pt} ε=0,1, \] driven by the same multidimensional martingale Y=(Yᵏ) and by different increasing processes V₀ʳ, V₁ʳ, r=1,2,…,d₁, where Lᵣ and Mᵏ are second-and first-order partial differential operators and ∘ stands for the Stratonovich differential. We estimate the moments of the supremum in t of the Sobolev norms of u₁(t)-u₀(t) in terms of the supremum of the differences |Vʳ₀ₜ-Vʳ₁ₜ|. Hence, we obtain moment estimates for the error of a multistage splitting-up method for stochastic PDEs, in particular, for the equation of the unnormalized conditional density in nonlinear filtering.