NDLI: Random Substitution of Time in Strong Markov Processes

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Random Substitution of Time in Strong Markov Processes

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Volkonskii, V. A.
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 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.
Starting Page	310
Ending Page	326
Page Count	17
File Format	PDF
ISSN	0040585X
DOI	10.1137/1103025
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	3
Volume Number	3
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-17
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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