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Zbl 1205.60130
Jentzen, A.; Kloeden, P.E.
The numerical approximation of stochastic partial differential equations. (English)
[J] Milan J. Math. 77, 205-244 (2009). ISSN 1424-9286; ISSN 1424-9294/e

Summary: 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.

MSC 2000:: *60H35 Computational methods for stochastic equations
60H15 Stochastic partial differential equations
60H15 Stochastic partial differential equations
35R60 PDE with randomness
60H35 Computational methods for stochastic equations
65C30 Stochastic differential and integral equations

Keywords: stochastic partial differential equations; stochastic ordinary differential equations; stochastic Taylor expansions; higher order numerical schemes; strong convergence; pathwise convergence

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Zentralblatt MATH Berlin [Germany]