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.
Publié le : 1977-10-14
Classification:
31-00,
Markov system,
additive functional,
central projection,
60J55,
60G05
@article{1176995711,
author = {Dynkin, E. B.},
title = {Markov Systems and Their Additive Functionals},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 653-677},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995711}
}
Dynkin, E. B. Markov Systems and Their Additive Functionals. Ann. Probab., Tome 5 (1977) no. 6, pp. 653-677. http://gdmltest.u-ga.fr/item/1176995711/