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We consider Markov processes in continuous time with state space $\posint^N$and provide two sufficient conditions and one necessary condition for theexistence of moments $E(\|X(t)\|^r)$ of all orders $r \in \nat$ for all $t \geq0$. The sufficient conditions also guarantee an exponential in time growthbound for the moments. The class of processes studied have finitely many stateindependent jumpsize vectors $\nu_1,\dots,\nu_M$. This class of processes arisenaturally in many applications such as stochastic models of chemical kinetics,population dynamics and queueing theory for example. We also provide anecessary and sufficient condition for stochiometric boundedness of species interms of $\nu_j$.

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