Damped jump-telegraph processes

TL;DR

This paper analyzes a Markov modulated jump process, deriving distribution and moment equations, and characterizing martingale distributions based on jump and velocity regimes, contributing to stochastic process theory.

Contribution

It introduces a novel framework for Markov modulated jump processes with state-dependent jump amplitudes and velocities, providing new equations and characterizations.

Findings

Derived distribution and moment equations for the process

Characterized martingale distributions in terms of observable regimes

Provided insights into the process dynamics and regime dependencies

Abstract

We study a one-dimensional Markov modulated random walk with jumps. It is assumed that amplitudes of jumps as well as a chosen velocity regime are random and depend on a time spent by the process at a previous state of the underlying Markov process. Equations for the distribution and equations for its moments are derived. We characterise the martingale distributions in terms of observable proportions between jump and velocity regimes.