Time-changed Poisson processes |
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Authors: | A. Kumar Erkan Nane P. Vellaisamy |
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Affiliation: | aDepartment of Mathematics, Indian Institute of Technology Bombay, Mumbai-400076, India;bDepartment of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA |
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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. |
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Keywords: | Hitting times Inverse Gaussian process Time-changed process, subordination Tempered stable processes Difference&ndash differential equation |
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