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 Zap. Nauchn. Sem. POMI, 2017, Volume 466, Pages 313–330 (Mi znsl6557)

On unattainable boundaries of a diffusion process range of values: semi-Markov approach

B. P. Harlamov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: One-dimensional homogeneous semi-Markov processes of diffusion type are considered. A transition function of such a process satisfy an ordinary second order differential equation. It is supposed that the process does not break and has no any interval of constancy. Under these conditions the Dirihlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1, and 0 on the boundaries of the interval. A criterion for the left boundary of the interval to be unattainable is derived, and for corresponding values 0, and 1 a criterion for the right boundary of the interval to be unattainable is derived. This criterion being applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.

Key words and phrases: ordinary differential equation, stochastically differential equation, diffusion Markov process, semi-Markov process of diffusion type, semi-Markov transition functions, unreachable edges of an interval, criterion for edges to be unreachable.

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Document Type: Article
UDC: 519.2

Citation: B. P. Harlamov, “On unattainable boundaries of a diffusion process range of values: semi-Markov approach”, Probability and statistics. Part 26, Zap. Nauchn. Sem. POMI, 466, POMI, St. Petersburg, 2017, 313–330

Citation in format AMSBIB
\Bibitem{Har17} \by B.~P.~Harlamov \paper On unattainable boundaries of a~diffusion process range of values: semi-Markov approach \inbook Probability and statistics. Part~26 \serial Zap. Nauchn. Sem. POMI \yr 2017 \vol 466 \pages 313--330 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6557} 

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This publication is cited in the following articles:
1. B. P. Kharlamov, “Ob integrale ot diffuzionnogo protsessa so znacheniyami na intervale s nedostizhimymi granitsami: polumarkovskii podkhod”, Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, POMI, SPb., 2018, 233–240
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