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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 2, Pages 240–259 (Mi tvp619)

On stochastic processes defined by differential equations

R. Z. Khas'minskii

Moscow

Abstract: Let the function $X_\varepsilon(\tau,\omega)$ be the solution of the problem (1.3). The main results of this paper are the following theorems.
Theorem 1. {\it If the function $F$ satisfies conditions (1.1), (1.2) and (1.4) the stochastic process $X_\varepsilon(\tau,\omega)$ has the following asymptotic behaviour
$$\sup_{0\le\tau\le\tau_0}\mathbf M|X_\varepsilon(\tau,\omega)-x^0(\tau)|\to0\quad(\varepsilon\to0),$$
where $x^0(\tau)$ is the solution of the problem} (1.5).
Theorem 2. {\it If $F$ satisfies conditions (3.1)–(3.4) and $\varepsilon\to0$ $n$-order distributions of the stochastic process $Y^{(\varepsilon)}(\tau,\omega)=\varepsilon^{-1/2}(X^{(\varepsilon)}(\tau,\omega)-x^0(\tau))$ approach those of the Gaussian Markov process} (3.6), (3.7).
In addition some applications of these theorems to problems of nonlinear mechanics are considered.

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English version:
Theory of Probability and its Applications, 1966, 11:2, 211–228

Bibliographic databases:

Citation: R. Z. Khas'minskii, “On stochastic processes defined by differential equations”, Teor. Veroyatnost. i Primenen., 11:2 (1966), 240–259; Theory Probab. Appl., 11:2 (1966), 211–228

Citation in format AMSBIB
\Bibitem{Kha66} \by R.~Z.~Khas'minskii \paper On stochastic processes defined by differential equations \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 2 \pages 240--259 \mathnet{http://mi.mathnet.ru/tvp619} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=203788} \zmath{https://zbmath.org/?q=an:0168.16002} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 2 \pages 211--228 \crossref{https://doi.org/10.1137/1111018} 

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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. M. I. Freidlin, “The averaging principle and theorems on large deviations”, Russian Math. Surveys, 33:5 (1978), 117–176
2. Yu. N. Barabanenkov, “Cauchy problem for stochastic Liouville equation with randomly variable Hamiltonian of perturbations in the form of a bounded operator”, Theoret. and Math. Phys., 42:1 (1980), 66–73
3. Yu. M. Kabanov, S. M. Pergamenshchikov, “Singular perturbations of stochastic differential equations”, Math. USSR-Sb., 71:1 (1992), 15–27
4. S. M. Pergamenshchikov, “Asymptotic expansions for a model with distinguished “fast” and “slow” variables, described by a system of singularly perturbed stochastic differential equations”, Russian Math. Surveys, 49:4 (1994), 1–44
5. Kifer Y., “Averaging principle for fully coupled dynamical systems and large deviations”, Ergodic Theory and Dynamical Systems, 24:3 (2004), 847–871
6. Bakhtin V., Kifer Y., “Diffusion approximation for slow motion in fully coupled averaging”, Probability Theory and Related Fields, 129:2 (2004), 157–181
7. Samoilenko A.M., Makhmudov N.I., Stanzhitskii A.N., “Averaging method and two–sided bounded solutions of ito stochastic systems”, Differential Equations, 43:1 (2007), 56–68
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9. N. O. Amelina, O. N. Granichin, A. L. Fradkov, “The method of averaged models for discrete-time adaptive systems”, Autom. Remote Control, 80:10 (2019), 1755–1782
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