A continuous semi-Markov process with values in a closed interval is considered. This process coincides with a Markov diffusion process inside the interval. Thus, violation of the Markov property is only possible at the boundary of the interval. We prove a sufficient condition under which a semi-Markov process is Markov. We show that, in addition to Markov processes with instantaneous reflection from the boundary of the interval. there exists a class of Markov processes with delayed reflection from the boundary. Such a process has a positive average measure of time at which its trajectory belongs to the boundaries. This gives a different proof of a similar result by Gikhman and Skorokhod of 1968. Bibliography: 5 titles.
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R. M. Blumenthal and R. K. Getoor, Markov Process and Potential Theory, Academic Press, New York (1968).
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Kiev (1968).
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B. P. Harlamov, “Diffusion process with delay at the ends of a segment,” Zap. Nauchn. Semin. POMI, 351, 284–297 (2007).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 368, 2009, pp. 243–267.
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Harlamov, B.P. On Markov diffusion processes with delayed reflection from boundaries of a segment. J Math Sci 167, 574–587 (2010). https://doi.org/10.1007/s10958-010-9945-6
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DOI: https://doi.org/10.1007/s10958-010-9945-6