Milan Journal of Mathematics

, Volume 77, Issue 1, pp 205–244 | Cite as

The Numerical Approximation of Stochastic Partial Differential Equations

Article

Abstract

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.

Mathematics Subject Classification (2000)

Primary 60H35 Secondary 60H15 

Keywords

Stochastic partial differential equations stochastic ordinary differential equations stochastic Taylor expansions higher order numerical schemes strong convergence pathwise convergence 

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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Institut für MathematikJohann Wolfgang Goethe-UniversitätFrankfurt am MainGermany

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