Explicit distributions and time-changed dynamics of the two-state Markov modulated Poisson process

Abstract

This paper investigates the two-state Markov modulated Poisson process Nt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_t$$\end{document}, also known as the switched Poisson process. Despite its wide range of applications, a comprehensive characterization of the probability law of Nt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_t$$\end{document} has not been fully addressed in the literature. We contribute to filling this gap by deriving explicit closed-form expressions for the probability distribution and by analyzing its asymptotic behavior. These results also establish a rigorous framework for extensions to more complex dynamics. Specifically, we investigate the time-changed switched Poisson process driven by a subordinator, focusing particularly on the Poisson process and on the inverse stable subordinator. For the latter case, we establish a connection with the fractional Poisson process. Our findings bridge Markov modulated Poisson process theory with fractional dynamics.