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.