Fractional Generalizations of the Compound Poisson Process

This paper introduces the Generalized Fractional Compound Poisson Process (GFCPP), which claims to be a unified fractional version of the compound Poisson process (CPP) that encom- passes existing variations as special cases. We derive its distributional properties, generalized fractional differential equations, and martingale properties. Some results related to the govern- ing differential equation about the special cases of jump distributions, including exponential, Mittag-Leffler, Bernst ́ein, discrete uniform, truncated geometric, and discrete logarithm. Some of processes in the literature such as the fractional Poisson process of order k, P ́olya-Aeppli process of order k, and fractional negative binomial process becomes the special case of the GFCPP. Classification based on arrivals by time-changing the compound Poisson process by the inverse tempered and the inverse of inverse Gaussian subordinators are studied. Finally, we present the simulation of the sample paths of the above-mentioned processes.