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On the Rate of Convergence of Splitting-up Approximations for SPDEs

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Stochastic Inequalities and Applications

Part of the book series: Progress in Probability ((PRPR,volume 56))

Abstract

We consider the stochastic PDE

$$du(t,x) = (Lu(t,x) + f(t,x))dt + ({{M}_{k}}u(t) + {{g}_{k}}(t,x)) \circ d{{W}^{k}},$$

where L and M k are second and first order partial differential operators, W is a multi-dimensional Wiener process, and ‘o’ stands for the Stratonovich differential. We approximate the solution by the splitting-up method (method of alternative directions) and we estimate the error in terms of Sobolev’s norm W m p (ℝd), and hence also in the supremum norm. We also show that our rate of convergence estimate is sharp.

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh, EH9 3JZ, UK

    István Gyöngy

  2. 127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455, USA

    Nicolai Krylov

Authors
  1. István Gyöngy
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  2. Nicolai Krylov
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Editor information

Editors and Affiliations

  1. Department of Mathematics, U-3009, University of Connecticut, Storrs, CT, 06268, USA

    Evariste Giné

  2. Laboratoire d’Analyse et de Mathématiques Appliquées CNRS UMR 8050, Université de Paris XII, 94010, Créteil Cedex, France

    Christian Houdré

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, 30332, USA

    Christian Houdré

  4. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007, Barcelona, Spain

    David Nualart

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© 2003 Springer Basel AG

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Gyöngy, I., Krylov, N. (2003). On the Rate of Convergence of Splitting-up Approximations for SPDEs. In: Giné, E., Houdré, C., Nualart, D. (eds) Stochastic Inequalities and Applications. Progress in Probability, vol 56. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8069-5_17

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  • DOI: https://doi.org/10.1007/978-3-0348-8069-5_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9428-9

  • Online ISBN: 978-3-0348-8069-5

  • eBook Packages: Springer Book Archive

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