Time-Changed Poisson Processes

Abstract

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.

18 pages

References in corpus (3)

• Brownian subordinators and fractional Cauchy problems
• Fokker-Planck-Kolmogorov equations associated with SDEs driven by time-changed fractional Brownian motion
• Higher order PDE's and iterated Processes

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• Compositions of Poisson and Gamma processes
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• Population processes sampled at random times
• Bivariate Tempered Space-Fractional Poisson Process and Shock Models
• Trajectory composition of Poisson time changes and Markov counting systems
• On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators
• Time-changed Poisson processes of order $k$
• Multivariate fractional Poisson processes and compound sums
• Some Time-changed fractional Poisson processes
• Asymptotic results for a multivariate version of the alternative fractional Poisson process