On the splitting-up method and stochastic partial differential equations

István Gyöngy, Nicolai Krylov

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

Abstract

We consider two stochastic partial differential equations duε(t) = (Lruε(t) + fr(t))dVεtr + (Mkuε(t) + gk(t)) ο dYtk, ε = 0, 1, driven by the same multidimensional martingale Y = (Yk) and by different increasing processes V0r, V1r, r = 1, 2,..., d1, where Lr and Mk 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 u1(t) - u0(t) in terms of the supremum of the differences |V0tr - V1tr|. 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.

Original languageEnglish (US)
Pages (from-to)564-591
Number of pages28
JournalAnnals of Probability
Volume31
Issue number2
DOIs
StatePublished - Apr 2003

Keywords

  • Splitting-up
  • Stochastic partial differential equations

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