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Teor. Veroyatnost. i Primenen., 1974, Volume 19, Issue 3, Pages 583–588 (Mi tvp2929)  

This article is cited in 3 scientific papers (total in 3 papers)

Short Communications

Approximate integration of stochastic differential equations

G. N. Mil'shtein

Moscow

Abstract: For the Ito equation
$$ dX=a(t,X) dt+\sigma(t,X) dw,\quad X(t_0)=x,\quad t_0\le t\le t_0+T $$
($w(t)$ is a standard Wiener process) the following approximation is proposed:
\begin{gather*} \overline X(t_0)=X(t_0),\quad\overline X(t_0+(k+1)h)=
=\overline X(t_0+kh)+\overline\sigma w_{k+1}+(\overline a-\frac12\overline\sigma\frac{\overline\partial\sigma}{\partial x})h+\frac12\overline\sigma\frac{\overline\partial\sigma}{\partial x}w_{k+1}^2 \end{gather*}
where $h=T/m$; $k=0,1,…,m-1$; $w_1,…,w_m$ are independent normal $N(0,h)$ variables. Here the stroke means that the corresponding function is computed at point $(t_0+kh,X(t_0+kh))$.
It is shown that $\mathbf M(X(t_0+T)-\overline X(t_0+T))^2=O(h^2)$.
The results are generalized to systems of stochastic differential equations.
Possibilities of improving the accuracy of the approximation are discussed.

Full text: PDF file (336 kB)

English version:
Theory of Probability and its Applications, 1975, 19:3, 557–562

Bibliographic databases:

Received: 23.09.1973

Citation: G. N. Mil'shtein, “Approximate integration of stochastic differential equations”, Teor. Veroyatnost. i Primenen., 19:3 (1974), 583–588

Citation in format AMSBIB
\Bibitem{Mil74}
\by G.~N.~Mil'shtein
\paper Approximate integration of stochastic differential equations
\jour Teor. Veroyatnost. i Primenen.
\yr 1974
\vol 19
\issue 3
\pages 583--588
\mathnet{http://mi.mathnet.ru/tvp2929}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=356225}
\zmath{https://zbmath.org/?q=an:0314.60039}
\transl
\jour Theory Probab. Appl.
\yr 1975
\vol 19
\issue 3
\pages 557--562
\crossref{http://dx.doi.org/10.1137/1119062}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Г. Н. Мильштейн, Н. Ф. Рыбкина, “Алгоритм метода блуждания по малым эллипсоидам для решения общей задачи Дирихле”, Ж. вычисл. матем. и матем. физ., 33:5 (1993), 704–725  mathnet  mathscinet  zmath; G. N. Mil'shtein, N. F. Rybkina, “An algorithm for random walks over small ellipsoids for solving the general Dirichlet problem”, Comput. Math. Math. Phys., 33:5 (1993), 631–647  isi
    2. Ю. С. Мишура, Г. М. Шевченко, “Аппроксимационные схемы для стохастических дифференциальных уравнений в гильбертовом пространстве”, ТВП, 51:3 (2006), 476–495  mathnet  crossref  mathscinet  zmath; Yu. S. Mishura, G. M. Shevchenko, “Approximation schemes for stochastic differential equations in Hilbert space”, Theory Probab. Appl., 51:3 (2007), 442–458  crossref  isi
    3. Jentzen A., Roeckner M., “A Milstein Scheme For Spdes”, Found. Comput. Math., 15:2 (2015), 313–362  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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