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Solving parabolic stochastic partial differential equations via averaging over characteristics

Authors: G. N. Milstein and M. V. Tretyakov
Journal: Math. Comp. 78 (2009), 2075-2106
MSC (2000): Primary 65C30, 60H15, 60H35, 60G35
Published electronically: March 6, 2009
MathSciNet review: 2521279
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Abstract: The method of characteristics (the averaging over the characteristic formula) and the weak-sense numerical integration of ordinary stochastic differential equations together with the Monte Carlo technique are used to propose numerical methods for linear stochastic partial differential equations (SPDEs). Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. A variance reduction technique for the Monte Carlo procedures is considered. Layer methods for linear and semilinear SPDEs are constructed and the corresponding convergence theorems are proved. The approach developed is supported by numerical experiments.

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Additional Information

G. N. Milstein
Affiliation: Ural State University, Lenin Str. 51, 620083 Ekaterinburg, Russia

M. V. Tretyakov
Affiliation: Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom

Keywords: Probabilistic representations of solutions of stochastic partial differential equations, numerical integration of stochastic differential equations, Monte Carlo technique, mean-square and almost sure convergence, layer methods.
Received by editor(s): May 30, 2007
Received by editor(s) in revised form: November 3, 2008
Published electronically: March 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society