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

We consider time homogeneous Markov processes in continuous time with state space $\mathbb {Z}_{+}^{N}$ and provide two sufficient conditions and one necessary condition for the existence of moments $E(\|X(t)\|^r)$ of all orders $r \in \mathbb {N}$ for all $t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied has finitely many state independent jumpsize vectors $\nu _1,\dots ,\nu _M$. This class of processes arises naturally in many applications such as stochastic models of chemical kinetics, population dynamics and epidemiology for example. We also provide a necessary and sufficient condition for stoichiometric boundedness of species in terms of $\nu _j$.