On the splitting-up method and stochastic partial differential equations
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István Gyöngy [1] ; Nicolai Krylov [2]
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[1] University of Edinburgh
University of Edinburgh
Reino Unido
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[2] University of Minnesota
University of Minnesota
City of Minneapolis, Estados Unidos
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 2, 2003, págs. 564-591
- Idioma: inglés
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Resumen
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=(Yk) and by different increasing processes Vr0, Vr1, r=1,2,…,d1, where Lr and Mk are second-and first-order partial differential operators and ∘ 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\break |Vr0t−Vr1t|. 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.
