On the splitting-up method and stochastic partial differential equations

We consider two stochastic partial differential equations du(epsilon)(t) = (L(r)u(epsilon)(t) + f(r)(t))dV(epsilont)(r) + (M(k)u(epsilon)(t) + g(k)(t)) odY(t)(k), epsilon = 0, 1, driven by the same multidimensional martingale Y = (Y-k) and by different increasing processes V-0(r), V-1(r), r = 1, 2,..., d(1), 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 u(1)(t) - u(0)(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.