Probability Surveys

On temporally completely monotone functions for Markov processes

Francis Hirsch and Marc Yor

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Abstract

Any negative moment of an increasing Lamperti process(Yt,t0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t0), the functions f such that the expectation of f(Yt) 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 Ben Saad and Janßen, 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.

Article information

Source
Probab. Surveys, Volume 9 (2012), 253-286.

Dates
First available in Project Euclid: 10 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ps/1336658304

Digital Object Identifier
doi:10.1214/11-PS179

Mathematical Reviews number (MathSciNet)
MR2947802

Zentralblatt MATH identifier
1245.60071

Subjects
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60G18: Self-similar processes

Keywords
Temporally completely monotone function completely excessive function completely superharmonic function Lamperti’s correspondence Lamperti process Markov process

Citation

Hirsch, Francis; Yor, Marc. On temporally completely monotone functions for Markov processes. Probab. Surveys 9 (2012), 253--286. doi:10.1214/11-PS179. https://projecteuclid.org/euclid.ps/1336658304


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