Convergence of a sequence of Markov chains to a diffusion type process
Authors:
G. L. Kulinich and A. V. Yershov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 78 (2009), 115-131
MSC (2000):
Primary 60H10; Secondary 34G10, 47A50, 47D06
DOI:
https://doi.org/10.1090/S0094-9000-09-00766-2
Published electronically:
August 4, 2009
MathSciNet review:
2446853
Full-text PDF Free Access
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Abstract: A random polygonal line constructed from a sequence of series of homogeneous Markov chains is considered under rather nonregular dependence of their local characteristics on a series number. Sufficient conditions are obtained for the weak convergence of a random polygonal line to a diffusion type process. The conditions are expressed explicitly in terms of local characteristics of the Markov chains.
References
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- Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic differential equations, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR 0346904
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- Grigorii L. Kulinich and Eugenii P. Kaskun, On the asymptotic behavior of solutions of one-dimensional Ito’s stochastic differential equations with singularity points, Proceedings of the Donetsk Colloquium on Probability Theory and Mathematical Statistics (1998), 1998, pp. 189–197. MR 2026628
- G. L. Kulīnīch, Necessary and sufficient conditions for the convergence of solutions of one-dimensional stochastic diffusion equations with irregular dependence of the coeffici, Teor. Veroyatnost. i Primenen. 27 (1982), no. 4, 795–802 (Russian, with English summary). MR 681473
- G. L. Kulinich, Asymptotic Analysis of Unstable Solutions of One-Dimensional Stochastic Differential Equations, Kyiv University, Kyiv, 2003. (Ukrainian)
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References
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, Nauka, Moscow, 1965; English transl., W. B. Saunders Co., Philadelphia, 1969. MR 0247660 (40:923)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, 1968; English transl., Springer-Verlag, Berlin–Heidelberg–New York, 1972. MR 0346904 (49:11625)
- N. V. Krylov, Controlled Diffusion Processes, Nauka, Moscow, 1977; English transl., Springer-Verlag, Berlin–New York, 1980. MR 508417 (80f:60046); MR 0601776 (82a:60062)
- G. L. Kulinich, Some limit theorems for a sequence of Markov chains, Teor. Veroyatnost. Matem. Statist. 12 (1975), 77–89; English transl. in Theory Probab. Math. Statist. 12 (1976), 79–92. MR 0397835 (53:1691)
- G. L. Kulinich and E. P. Kaskun, On the asymptotic behavior of solutions of one-dimensional Ito’s stochastic differential equations with singularity points, Theory Stoch. Process. 4(20) (1998), no. 1–2, 189–197. MR 2026628 (2004j:60122)
- G. L. Kulinich, Necessary and sufficient conditions for the convergence of solutions of one-dimensional stochastic differential equations with irregular dependence of the coefficients on a parameter, Teor. Veroyatnost. Primenen. 27 (1982), no. 4, 795–801; English transl. in Theory Probab. Appl. 27 (1982), no. 4, 856–862. MR 681473 (84g:60093)
- G. L. Kulinich, Asymptotic Analysis of Unstable Solutions of One-Dimensional Stochastic Differential Equations, Kyiv University, Kyiv, 2003. (Ukrainian)
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Additional Information
G. L. Kulinich
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
a_yershov@univ.kiev.ua
A. V. Yershov
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Keywords:
A sequence of series of Markov chains,
nonregular dependence of local characteristics of Markov chains on the number of a series,
a random polygonal line,
weak convergence,
stochastic differential equation,
diffusion type processes
Received by editor(s):
May 7, 2007
Published electronically:
August 4, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
