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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1958, Volume 3, Issue 3, Pages 332–350 (Mi tvp4938)

Random Substitution of Time in Strong Markov Processes

V. A. Volkonskii

Moscow

Abstract: The terminology and symbols are as in [7] and [1].
Let $x(t,\omega)$ be a homogeneous strong Markov process, and $\tau_t(\omega)$ be a random function not decreasing for increasing $t$. The process $y_t=x(\tau_t(\omega),\omega)$ is called a process obtained from $x_t(\omega)$ by means of a random substitution of time $\tau_t$.
The conditions sufficient for the process $y_t$ to be a Markov or a strong Markov process are formulated (Theorems 1 and 2).
In [1] it is shown that the infinitesimal operator $\mathrm A$ of $a$ Feller strong Markov process continuous on the right is a contraction of a certain operator $\mathfrak{a}$, which is called the extended operator. It is shown that if $x_t$ and $x(\tau _t)$ are Feller processes continuous on the right and $\tau _t$ is determined by equation (2), where $\varphi (x)>0$, and continuous, then their extended operator is $\mathfrak{a}$, where $\mathfrak{a}$ satisfies the equation $t=\varphi (x)\mathfrak{a}$ (Theorem 3).
In Theorem 4 and in its corollary it is shown that a one-dimensional homogeneous regular continuous strong Markov process may be obtained from a Wiener process by means of a random substitution of time and a monotone transformation of the segment.

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English version:
Theory of Probability and its Applications, 1958, 3:3, 310–326

Citation: V. A. Volkonskii, “Random Substitution of Time in Strong Markov Processes”, Teor. Veroyatnost. i Primenen., 3:3 (1958), 332–350; Theory Probab. Appl., 3:3 (1958), 310–326

Citation in format AMSBIB
\Bibitem{Vol58} \by V.~A.~Volkonskii \paper Random Substitution of Time in Strong Markov Processes \jour Teor. Veroyatnost. i Primenen. \yr 1958 \vol 3 \issue 3 \pages 332--350 \mathnet{http://mi.mathnet.ru/tvp4938} \transl \jour Theory Probab. Appl. \yr 1958 \vol 3 \issue 3 \pages 310--326 \crossref{https://doi.org/10.1137/1103025} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Baurdoux E.J. Kyprianou A.E. Ott C., “Optimal prediction for positive self-similar Markov?processes”, Electron. J. Probab., 21 (2016), 48
2. Doering L. Horvath B. Teichmann J., “Functional Analytic (Ir-) Regularity Properties of Sabr-Type Processes”, Int. J. Theor. Appl. Financ., 20:3 (2017), 1750013