On the splitting-up method and stochastic partial differential equations

We consider two stochastic partial differential equations du_ε(t) = (L_ru_ε(t) + f_r(t))dV_εt^r + (M_ku_ε(t) + g_k(t)) ο dY_t^k, ε = 0, 1, driven by the same multidimensional martingale Y = (Y^k) and by different increasing processes V₀^r, V₁^r, r = 1, 2,..., d₁, where L_r and M^k are second- and first-order partial differential operators and o 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 |V_0t^r - V_1t^r|. 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.