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Teoriya Veroyatnostei i ee Primeneniya, 2002, Volume 47, Issue 1, Pages 39–58
DOI: https://doi.org/10.4213/tvp2965
(Mi tvp2965)
 

This article is cited in 6 scientific papers (total in 6 papers)

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.
Received: 16.11.1999
English version:
Theory of Probability and its Applications, 2003, Volume 47, Issue 1, Pages 53–68
DOI: https://doi.org/S0040585X97979433
Bibliographic databases:
Document Type: Article
Language: Russian
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
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\paper The simplest random walks for the Dirichlet problem
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\pages 39--58
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\jour Theory Probab. Appl.
\yr 2003
\vol 47
\issue 1
\pages 53--68
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  • https://doi.org/10.4213/tvp2965
  • https://www.mathnet.ru/eng/tvp/v47/i1/p39
  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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