The Annals of Probability

Markov Systems and Their Additive Functionals

E. B. Dynkin

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Abstract

For certain classes of Markov systems (that is, stochastic systems which have Markov representations with transition and cotransition probabilities) considered by the author in previous papers, a correspondence was established between additive functionals of any such system and measures on a certain measurable space. We now prove analogous results for arbitrary Markov systems. Measures corresponding to the additive functionals are defined on a certain $\sigma$-algebra in the product space $R \times \Omega$ where $R$ is the real line and $\Omega$ is the sample space (we call it the central $\sigma$-algebra). The theory is applicable not only to traditional processes but also to a number of generalized stochastic processes introduced by Gelfand and Ito. A situation where the observations are performed over a random time interval and the measure $P$ can be infinite is considered in the concluding section. These generalizations are of special importance for the homogeneous case which will be treated in another publication.

Article information

Source
Ann. Probab., Volume 5, Number 5 (1977), 653-677.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995711

Digital Object Identifier
doi:10.1214/aop/1176995711

Mathematical Reviews number (MathSciNet)
MR451415

Zentralblatt MATH identifier
0379.60076

JSTOR
links.jstor.org

Subjects
Primary: 60J55: Local time and additive functionals
Secondary: 60G05: Foundations of stochastic processes

Keywords
31-00 Markov system additive functional central projection

Citation

Dynkin, E. B. Markov Systems and Their Additive Functionals. Ann. Probab. 5 (1977), no. 5, 653--677. doi:10.1214/aop/1176995711. https://projecteuclid.org/euclid.aop/1176995711


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