Time-Changed Poisson Processes

Details Time-Changed Poisson Processes Time-Changed Poisson Processes

2011-05-03

arxiv.org/abs/1105.0657v1

Statistics & Probability Letters. 81 (2011) 1899-1910

Mathematics

Probability

18 pages

Scientific paper

10.1016/j.spl.2011.08.002

We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index $0<\beta<1,$ when $\beta $ is a rational number. We then use this result to obtain the governing DDE for the mass function of Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. \cite{B-M-N} and Beghin et al. \cite{BO-1} in several directions.

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