Particular solutions of the Chapman—Kolmogorov equation for multi-dimensionalstate Markov process with continuous time
DOI:
https://doi.org/10.21638/11701/spbu01.2016.205Abstract
The task of its immediate solution without the linearization was set in 1932 by S. N. Bernstein and partially was solved in 1961 by O. V. Sarmanov as bilinear series. In 2007–2010 author found several partial solutions of the above equation in the form of both a series of the Sarmanov-type and an integral. It was assumed that the state space of a Markov process was one-dimensional. In the article three particular solutions are found as integrals for multi-dimensional-state Markov process. Results are illustrated with five examples, one of which shows that it is the solution of the original equation that does not have a probabilistic sense. Refs 8.Downloads
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References
Литература
1. Вероятность и математическая статистика. Энциклопедический словарь. М., 2003. 912 с.
2. Бернштейн С.Н. О зависимостях между случайными величинами // Собр. соч.: в 4 т. М.: Наука, 1964. Т. 4. С. 235-254.
3. Сарманов О.В. Исследование стационарных марковских процессов методом разложения по собственным функциям // Труды матем. ин-та АН СССР. Т. 60. М.: Наука, 1961. С. 238-259.
4. Мирошин Р.Н. О некоторых решениях интегрального уравнения Колмогорова-Чепмена // Вестник С-Петерб. ун-та. Сер. 1. 2007. Вып. 4. С. 22-29.
5. Мирошин Р.Н. О многократных интегралах специального вида // Матем. заметки. 2007. Т. 82, вып. 3. С. 401-410.
6. Бейтмен Г., Эрдейи А. Высшие трансцендентные функции / пер. с англ. М.: Наука, 1966. Т. 2. 295 с.
7. Бохнер С. Лекции об интегралах Фурье / пер. с англ. М.: ГИФМЛ, 1962. 360 с.
8. Уилкс С. Математическая статистика / пер. с англ. М.: Наука, 1967. 632 с.
References
1. Probability and Mathematical Statistics. Encyclopedic dictionary (Bol’shaya Rossiiskaya Entsiklopediya, Moskow, 2003) [in Russian].
2. Bernstein S.N., “On Dependencies between Random Values”, Collected works 4, 235–254 (Nauka, Moscow, 1964) [in Russian].
3. Sarmanov O.V., “Investigation of Stationary Markov Processes by the Method of Eigenfunction Expansion”, Tr. Mat. Inst. Steklova 60, 238–259 (1961) [in Russian].
4. Miroshin R.N., “On Some Solutions to the Chapman—Kolmogorov Integral Equation”, Vestnik St.Petersb. Univ. Math. 40, Issue 4, 253–259 (2007).
5. Miroshin R. N., “On Multiple Integrals of Special form”, Math. Notes 82, Issue 3, 357–365 (2007).
6. Bateman H., Erd´elyi A., Higher Transtendental Functions 2 (McGraw-Hill, New York-Toronto- London, 1953; Nauka, Moskow, 1970).
7. Bochner S., Lectures on Fourier Integrals (Princeton University Press, Princeton-New Jersey, 1959; GIFML, Moskow, 1962).
8. Wilks S. S. Mathematical Statistics (Wiley, New York, 1961; Nauka, Moskow, 1967).
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Published
2020-10-19
How to Cite
Miroshin, R. N. (2020). Particular solutions of the Chapman—Kolmogorov equation for multi-dimensionalstate Markov process with continuous time. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 3(2), 1. https://doi.org/10.21638/11701/spbu01.2016.205
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
