NDLI: On Strong Markov Processes

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On Strong Markov Processes

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Yushkevich, A. A.
Copyright Year	1957
Abstract	The problem of constructing a strong Markov process with a given measurable Markov transition function $p(s,x,t,\Gamma )$ is considered. The space X of possible states is supposed to be given together with the function $p(s,x,t,\Gamma )$.If it is required that the sample functions $x(t,\omega )$ be defined for each $\omega $ at all $t \in [0,\infty )$, then as shown by the examples described, the three following types of measurable transition functions exist: 1) every Markov process with a given transition function is a strong Markov process; 2) there are both strong Markov processes, and measurable not strong Markov processes with a given transition function; 3) there are no strong Markov processes with a given transition function. To avoid the third case, it is important to know, whether there exists a process with a given transition function whose sample functions are continuous from the right. Under some general assumptions about the space X, a process with such sample functions exists if for each $s \geqq 0$ and $\varepsilon \geqq 0$ uniformly in $x \in X$\[ {\text{(I)}}\qquad \mathop {\lim }\limits_{t \downarrow s} p(u,x,t,V_\varepsilon (x)) = 1,\quad s \leqq u \leqq t, \] where $V_\varepsilon (x)$ is the $\varepsilon $-neighborhood of the point x.For the discrete topology, every Markov process with sample functions continuous from the right is a strong Markov process. If for each $s \geqq 0$ uniformly in $x \in X$\[ {\text{(II)}} \qquad \mathop {\lim }\limits_{t \downarrow s} p(s,x,t,x) = 1, \] then the sample functions of the process can be assumed to be continuous from the right in the discrete topology. For the case of a finite number of states, condition (II) is also necessary for continuity of the sample functions from the right.If it is not required that the sample functions $x(t,\omega )$ be defined for each $\omega $ at all t, then in all known examples of measurable transition functions of type (3), one can obtain the strong Markov property, which is now defined in accordance with the new character of the sample functions. This can be done for every transition function of a time-homogeneous process with a denumerable number of states for which the condition (II) is true (without uniformity in x).
Starting Page	181
Ending Page	205
Page Count	25
File Format	PDF
ISSN	0040585X
DOI	10.1137/1102013
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	2
Volume Number	2
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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