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 Uspekhi Mat. Nauk, 1975, Volume 30, Issue 1(181), Pages 61–99 (Mi umn4127)

Markov representations of stochastic systems

E. B. Dynkin

Abstract: A great deal of research into the theory of random processes is concerned with the problem of constructing a process that has certain properties of regularity of the trajectories and has the same finite-dimensional probability distribution as a given stochastic process $x_t$. It is a complicated theory and one that is difficult to apply to those properties that we most need for the study of Markov processes (the strict Markov property, standardization, and the like.) The problem can be usefully reformulated. In an actual experiment we do not observe the state $x_t$ at a fixed instant $t$, but rather events that occupy certain time intervals. This is the motivation behind the Gel'fand–Itô theory of generalized random processes. Kolmogorov, in 1972, proposed an even more general concept of a stochastic process as a system of $\sigma$-algebras $\mathscr F(I)$ labelled by time intervals $I$. Developing this approach, we introduce the concept of a Markov representation $x_t$ of the stochastic system $\mathscr F(I)$ and prove the existence of regular representations. We construct two dual regular representations (the right and the left), which we then combine into a single Markov process by two methods, the “vertical” and the “horizontal” method. We arrive at a general duality theory, which provides a natural framework for the fundamental results on entry and exit spaces, excessive measures and functions, additive functionals, and others. The initial steps in the construction of this theory were taken in [6]. The note [5] deals with applications to additive functionals (detailed proofs are in preparation). We consider random processes defined in measurable spaces without any topology: the introduction of a reasonable topology allows of a certain arbitrariness. The relation between our definitions of regularity and more traditional properties stated in topological terms (continuity from the right, the existence of a limit from the left, etc.) are considered in the Appendix, which is written by S. E. Kuznetsov.

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English version:
Russian Mathematical Surveys, 1975, 30:1, 65–104

Bibliographic databases:

UDC: 519.24
MSC: 60G20, 60J57, 46E30, 60Exx, 60Fxx

Citation: E. B. Dynkin, “Markov representations of stochastic systems”, Uspekhi Mat. Nauk, 30:1(181) (1975), 61–99; Russian Math. Surveys, 30:1 (1975), 65–104

Citation in format AMSBIB
\Bibitem{Dyn75} \by E.~B.~Dynkin \paper Markov representations of stochastic systems \jour Uspekhi Mat. Nauk \yr 1975 \vol 30 \issue 1(181) \pages 61--99 \mathnet{http://mi.mathnet.ru/umn4127} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=402887} \zmath{https://zbmath.org/?q=an:0316.60019} \transl \jour Russian Math. Surveys \yr 1975 \vol 30 \issue 1 \pages 65--104 \crossref{https://doi.org/10.1070/RM1975v030n01ABEH001401} 

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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. Stephan Lawi, “Towards a Characterization of Markov Processes Enjoying the Time-Inversion Property”, J Theoret Probab, 21:1 (2008), 144
2. Xian-min Geng, Liang Li, “Markov process functionals in finance and insurance”, Appl Math Chin Univ, 24:1 (2009), 21
3. È. B. Vinberg, S. E. Kuznetsov, “Evgenii (Eugene) Borisovich Dynkin (obituary)”, Russian Math. Surveys, 71:2 (2016), 345–371
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