Abstract
A continuous semi-Markov process with a segment as the range of values is considered. This process coincides with a diffusion process inside the segment, i.e., up to the first hitting time of the boundary of the segment and at any time when the process leaves the boundary. The class of such processes consists of Markov processes with reflection at the boundaries (instantaneously or with a delay) and semi-Markov processes with intervals of constancy on some boundary. We derive conditions of existence of such a process in terms of a semi-Markov transition generating function on the boundary. The method of imbedded alternating renewal processes is applied to find a stationary distribution of the process. Bibliography: 3 titles.
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References
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Kiev (1968).
B. M. Shurenkov, Ergodic Markov Processes [in Russian], Moscow (1989).
B. P. Harlamov, Continuous Semi-Markov Processes, ISTE & Wiley (2008).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 284–297.
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Harlamov, B.P. Diffusion processes with delay at the endpoints of a segment. J Math Sci 152, 958–965 (2008). https://doi.org/10.1007/s10958-008-9114-3
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DOI: https://doi.org/10.1007/s10958-008-9114-3