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COUNTABLE STATE MARKOV PROCESSES: NON-EXPLOSIVENESS AND MOMENT FUNCTION

Published online by Cambridge University Press:  09 July 2015

F.M. Spieksma
F.M. Spieksma*
Affiliation:
Mathematics Institute, Leiden University, Niels Borhweg 1, 2333CA Leiden, The Netherlands Email: spieksma@math.leidenuniv.nl
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Abstract

The existence of a moment function satisfying a drift function condition is well known to guarantee non-explosiveness of the associated minimal Markov process (cf. [1,9]), under standard technical conditions. Surprisingly, the reverse is true as well for a countable space Markov process. We prove this result by showing that recurrence of an associated jump process, that we call the α-jump process, is equivalent to non-explosiveness. Non-explosiveness corresponds in a natural way to the validity of the Kolmogorov integral relation for the function identically equal to 1. In particular, we show that positive recurrence of the α-jump chain implies that all bounded functions satisfy the Kolmogorov integral relation. We present a drift function criterion characterizing positive recurrence of this α-jump chain.

Suppose that to a drift function V there corresponds another drift function W, which is a moment with respect to V. Via a transformation argument, the above relations hold for the transformed process with respect to V. Transferring the results back to the original process, allows to characterize the V-bounded functions that satisfy the Kolmogorov forward equation.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 4 , October 2015 , pp. 623 - 637
Copyright
Copyright © Cambridge University Press 2015 

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