An Increment-Type Set-Indexed Markov Property

Paul Balança ()
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Paul Balança: École Centrale Paris

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1271-1310

Abstract: Abstract We present and study a Markov property, named $$\mathcal C$$ C -Markov, adapted to processes indexed by a general collection of sets. This new definition fulfils one important expectation for a set-indexed Markov property: there exists a natural generalization of the concept of transition operator which leads to characterization and construction theorems of $$\mathcal C$$ C -Markov processes. Several usual Markovian notions, including Feller and strong Markov properties, are also developed in this framework. Actually, the $$\mathcal C$$ C -Markov property turns out to be a natural extension of the two-parameter $$*$$ ∗ -Markov property to the multiparameter and the set-indexed settings. Moreover, extending a classic result of the real-parameter Markov theory, sample paths of multiparameter $$\mathcal C$$ C -Feller processes are proved to be almost surely right-continuous. Concepts and results presented in this study are illustrated with various examples.

Keywords: Markov property; Multiparameter processes; Transition system; Set-indexed processes; 60G10; 60G15; 60G60; 60J25 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0555-y

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