In this paper, we introduce and study fractional versions of the Bell–Touchard process, Poisson-logarithmic process generalized Pólya–Aeppli process. The state probabilities these compound Poisson processes solve a system differential equations that involves Caputo derivative order 0<β<1. It is shown are limiting cases recently introduced namely, counting We obtain mean, variance, covaria...

The problem of controlling a compound Poisson process until it leaves an interval is considered. This type known as homing problem. To determine the value optimal control, we must solve nonlinear integro-differential equation. Exact and explicit solutions are obtained for two possible jumps size distributions.

The Poisson process is a stochastic counting process that arises naturally in a large variety of daily-life situations. We present a few definitions of the Poisson process and discuss several properties as well as relations to some well-known probability distributions. We further briefly discuss the compound Poisson process.

Ordinary differential equations(ODEs) with stochastic processes in their vector field, have lots of applications in science and engineering. The main purpose of this article is to investigate the numerical methods for ODEs with Wiener and Compound Poisson processes in more than one dimension. Ordinary differential equations with Ito diffusion which is a solution of an Ito stochastic differentia...