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Hirsch, Francis; Yor, Marc

On temporally completely monotone functions for Markov processes. (English) Zbl 1245.60071

Probab. Surv. 9, 253-286 (2012).
Summary: Any negative moment of an increasing Lamperti process \((Y_{t},t\geq 0)\) is a completely monotone function of \(t\). This property enticed us to study systematically for a given Markov process (\(Y_{t},t\geq 0\)) the functions \(f\) such that the expectation of \(f(Y_{t})\) is a completely monotone function of \(t\). We call these functions temporally completely monotone (for \(Y\)). Our description of these functions is deduced from the analysis made by H. Ben Saad and K. Janßen [“Bernstein’s theorem for completely excessive measures”, Nagoya Math. J. 119, 133–141 (1990), euclid:nmj/1118782040] in a general framework of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when \(Y\) is a Lévy process, a Bessel process, or an increasing Lamperti process.

Cited in 1 Review
Cited in 5 Documents

MSC:

60J45 Probabilistic potential theory
60J25 Continuous-time Markov processes on general state spaces
60J35 Transition functions, generators and resolvents
60G18 Self-similar stochastic processes

Keywords:

temporally completely monotone function; completely excessive function; completely superharmonic function; Lamperti’s correspondence; Lamperti process; Markov process
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Full Text: DOI Euclid

References:

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