On the splitting-up method and stochastic partial differential equations

Abstract

We consider two stochastic partial differential equations \[ du_{\varepsilon}(t)= (L_ru_{\varepsilon}(t)+f_{r}(t)) \,dV_{\varepsilon t}^r+(M_{k}u_{\varepsilon}(t)+g_k(t))\, \circ dY_t^k, \qquad\hspace*{-5pt} \varepsilon=0,1, \] driven by the same multidimensional martingale $Y=(Y^k)$ and by different increasing processes $V_{0}^r$, $V_1^r$, $r=1,2,\ldots,d_1$, where $L_r$ and $M^k$ are second-and first-order partial differential operators and $\circ$ 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\break $|V^r_{0t}-V^{r}_{1t}|$. 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.