Generalized Iterated Poisson Process and Applications

In this paper, we consider the composition of a homogeneous Poisson process with an independent time-fractional Poisson process. We call this composition the generalized iterated Poisson process (GIPP). The probability law in terms of the fractional Bell polynomials, governing fractional differential equations, and the compound representation of the GIPP are obtained. We give explicit expressions for mean and covariance and study the long-range dependence property of the GIPP. It is also shown that the GIPP is over-dispersed. Some results related to first-passage time distribution and the hitting probability are also examined. We define the compound and the multivariate versions of the GIPP and explore their main characteristics. Further, we consider a surplus model based on the compound version of the iterated Poisson process (IPP) and derive several results related to ruin theory. Its applications using the Poisson–Lindley and the zero-truncated geometric distributions are also provided. Finally, simulated sample paths for the IPP and the GIPP are presented.