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 Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 1, Pages 39–58 (Mi tvp2965)

The simplest random walks for the Dirichlet problem

G. N. Mil'shteina, M. V. Tretyakovb

a Ural State University
b Mathematics Department, University of Leicester

Abstract: The Dirichlet problem for both parabolic and elliptic equations is considered. A solution of the corresponding characteristic system of stochastic differential equations is approximated in the weak sense by a Markov chain. If a state of the chain comes close to the boundary of the domain in which the problem is considered, then in the next step the chain either stops on the boundary or goes inside the domain with some probability due to an interpolation law. An approximate solution of the Dirichlet problem has the form of expectation of a functional of the chain trajectory. This makes it possible to use the Monte Carlo technique. The proposed methods are the simplest ones because they are based on the weak Euler approximation and linear interpolation. Convergence theorems, which give accuracy orders of the methods, are proved. Results of some numerical tests are presented.

Keywords: Dirichlet problem for parabolic and elliptic equations, probabilistic representations, weak approximation of solutions of stochastic differential equations, Markov chains, random walks.

DOI: https://doi.org/10.4213/tvp2965

Full text: PDF file (1933 kB)

English version:
Theory of Probability and its Applications, 2003, 47:1, 53–68

Bibliographic databases:

Citation: G. N. Mil'shtein, M. V. Tretyakov, “The simplest random walks for the Dirichlet problem”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 39–58; Theory Probab. Appl., 47:1 (2003), 53–68

Citation in format AMSBIB
\Bibitem{MilTre02} \by G.~N.~Mil'shtein, M.~V.~Tretyakov \paper The simplest random walks for the Dirichlet problem \jour Teor. Veroyatnost. i Primenen. \yr 2002 \vol 47 \issue 1 \pages 39--58 \mathnet{http://mi.mathnet.ru/tvp2965} \crossref{https://doi.org/10.4213/tvp2965} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1978694} \zmath{https://zbmath.org/?q=an:1038.60066} \transl \jour Theory Probab. Appl. \yr 2003 \vol 47 \issue 1 \pages 53--68 \crossref{https://doi.org/S0040585X97979433} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000183800400004} 

• http://mi.mathnet.ru/eng/tvp2965
• https://doi.org/10.4213/tvp2965
• http://mi.mathnet.ru/eng/tvp/v47/i1/p39

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Milstein G.N., Tretyakov M.V., “Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach”, IMA Journal of Numerical Analysis, 21:4 (2001), 887–917
2. Buchmann F.M., “Simulation of stopped diffusions”, Journal of Computational Physics, 202:2 (2005), 446–462
3. Goldberg M.J., Kim S., “Applications of some formulas for finite Markov chains”, Appl Comput Harmon Anal, 30:1 (2011), 37–46
4. Milstein G.N., Tretyakov M.V., “Solving the Dirichlet problem for Navier–Stokes equations by probabilistic approach”, BIT Numerical Mathematics, 52:1 (2012), 141–153
5. Chigansky P. Klebaner F.C., “The Euler-Maruyama Approximation for the Absorption Time of the Cev Diffusion”, Discrete Contin. Dyn. Syst.-Ser. B, 17:5 (2012), 1455–1471
6. Bernal F., Acebron J.A., “A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations”, Commun. Comput. Phys., 20:3 (2016), 703–732