Moment growth bounds on continuous time Markov processes on non-negative integer lattices

TL;DR

This paper establishes conditions for the existence and growth bounds of moments in continuous-time Markov processes on non-negative integer lattices, with applications in chemical kinetics, population dynamics, and queueing theory.

Contribution

It provides two sufficient and one necessary condition for moments' existence, along with exponential growth bounds, for Markov processes with finitely many state-independent jumps.

Findings

Conditions for moments' existence and growth bounds.

Necessary and sufficient condition for species' boundedness.

Applicable to models in chemistry, biology, and queueing systems.

Abstract

We consider Markov processes in continuous time with state space $\posint^N$ and provide two sufficient conditions and one necessary condition for the existence of moments $E(\|X(t)\|^r)$ of all orders $r \in \nat$ for all $t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied have finitely many state independent jumpsize vectors $\nu_1,\dots,\nu_M$. This class of processes arise naturally in many applications such as stochastic models of chemical kinetics, population dynamics and queueing theory for example. We also provide a necessary and sufficient condition for stochiometric boundedness of species in terms of $\nu_j$.