Time-changed Poisson processes

A. Kumar, Erkan Nane and P. Vellaisamy

Statistics & Probability Letters, 2011, vol. 81, issue 12, 1899-1910

Abstract: We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDEs. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDEs 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<β<1, when β is a rational number. We then use this result to obtain the governing DDE for the mass function of the Poisson process time-changed by the tempered stable subordinator. Our results extend and complement the results in Baeumer et al. (2009) and Beghin and Orsingher (2009) in several directions.

Keywords: Hitting times; Inverse Gaussian process; Time-changed process, subordination; Tempered stable processes; Difference–differential equation (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (7)

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DOI: 10.1016/j.spl.2011.08.002

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