Behavior of an Almost Semicontinuous Poisson Process on a Markov Chain Upon Attainment of A Level

TL;DR

This paper analyzes almost semi-continuous Poisson processes on finite Markov chains, deriving their moment generating functions for maximum levels and recovery times, including modified processes with two-step negative jump rates.

Contribution

It provides new representations for key process characteristics and investigates modified processes with complex jump behaviors.

Findings

Derived moment generating functions for maximum and recovery times.

Analyzed processes with two-step negative jump rates.

Provided insights into process behavior on finite Markov chains.

Abstract

We consider the almost semi-continuous processes defined on a finite Markov chain. The representation of the moment generating functions for the absolute maximum after achievement positive level and for the recovery time are obtained. Modified processes with two-step rate of negative jumps are investigated.