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Theory of Probability and Mathematical Statistics

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Extended Poisson equation for weakly ergodic Markov processes

Authors: A. Yu. Veretennikov and A. M. Kulik
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
Journal: Theor. Probability and Math. Statist. 85 (2012), 23-39
MSC (2010): Primary 60H10, 60J10
Published electronically: January 11, 2013
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Abstract: Solvability conditions for a Poisson equation with an extended generator of a general Markov process are obtained. The predictable part in the Doob-Meyer decomposition is described for a process of the form $ g(X(t), Y(t))$, where $ Y$ is a solution of a stochastic equation with the coefficients depending on $ X$ and where the function $ g=g(x,y)$ is defined as a family of solutions of the Poisson equation.

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Additional Information

A. Yu. Veretennikov
Affiliation: School of Mathematics, University of Leeds, LS2 9JT, Leeds, United Kingdom; Institute of Information Transmission Problems, Bol’shoi Karetny Street 19, 127994, Moscow, Russia

A. M. Kulik
Affiliation: Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 01601 Kyiv–4, Ukraine

Keywords: Markov process, extended generator, Poisson equation, Doob–Meyer decomposition
Received by editor(s): September 6, 2011
Published electronically: January 11, 2013
Additional Notes: This research was partially supported by the State Fund for Fundamental Researches, project $Φ$40.1/023
This paper is based on the talk presented at the International Conference “Modern Stochastics: Theory and Applications II” held on September 7–11, 2010, at Kyiv National Taras Shevchenko University and dedicated to the anniversaries of the prominent Ukrainian scientists Anatoliĭ Skorokhod, Vladimir Korolyuk, and Igor Kovalenko
Article copyright: © Copyright 2013 American Mathematical Society

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