stipulation , and say is stable or totally stable to mean , i.e., every entry is real valued.) Note that the row sums of are 0: or more succinctly, . This situation contrasts with the situation for discrete-time Markov chains, where all row sums of the transition matrix equal unity. Now, let such that is -measurable. There are three equivalent ways to define being Markov with initial distribution and rate matrix : via transition probabilities or via the jump chain and holding times. As a prelude to a transition-probability definition, we first motivate the definition of a regular rate matrix. We will use the transition rate matrix to specify the dynamics of the Markov chain by means of generating a collection of transition matrices on (), via the following theorem. We say is regular to mean that we do have uniqueness for the above system, i.e., that there exists exactly one solution. We say is irregular to mean is not regular. If is finite, then there is exactly one solution, namely and hence is regular. Otherwise, is infinite, and there exist irregular transition rate matrices on . If is regular, then for the unique solution , for each , will be a stochastic matrix. We will assume is regular from the beginning of the following subsection up through the end of this section, even though it is conventional to not include this assumption. (Note for the expert: thus we are not defining continuous-time Markov chains in general but only non-explosive continuous-time Markov chains.) Transition-probability definition Let be the (unique) solution of the system (). (Uniqueness guaranteed by our assumption that is regular.) We say is Markov with initial distribution and rate matrix to mean: for any nonnegative integer , for all such that for all Using induction and the fact that we can show the equivalence of the above statement containing () and the following statement: for all and for any nonnegative integer , for all such that for all such that (it follows that ), It follows from continuity of the functions () that the trajectory is almost surely right continuous (with respect to the discrete metric on ): there exists a -null set such that . Jump-chain/holding-time definition Sequences associated to a right-continuous function Let be right continuous (when we equip with the discrete metric). Define let be the holding-time sequence associated to , choose and let be