Annals of Probability

On the splitting-up method and stochastic partial differential equations

István Gyöngy and Nicolai Krylov

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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.

Article information

Source
Ann. Probab., Volume 31, Number 2 (2003), 564-591.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1048516528

Digital Object Identifier
doi:10.1214/aop/1048516528

Mathematical Reviews number (MathSciNet)
MR1964941

Zentralblatt MATH identifier
1028.60058

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 65M12: Stability and convergence of numerical methods 65M15: Error bounds 93E11: Filtering [See also 60G35]

Keywords
Stochastic partial differential equations splitting-up

Citation

Gyöngy, István; Krylov, Nicolai. On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31 (2003), no. 2, 564--591. doi:10.1214/aop/1048516528. https://projecteuclid.org/euclid.aop/1048516528


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  • MINNEAPOLIS, MINNESOTA 55455 E-MAIL: kry lov@math.umn.edu