[Math] Transition probability in Continuous Time Markov Chain (CTMC)

Question

I know that for a CTMC, the transition matrix $P(t)=e^{tQ}$, where $Q$ is the infinitesimal generator matrix of the irriducible CTMC.

My question is how do I deal with situations or problems that ask me to compute the probability of going from state $i$ to state $j$ after $s$ time? This is rather straight forward for a Discrete Time Markov Chain (DTMC), where all I have to do is get $P^s_{i,j}$, but I have no idea what to do for the CTMC case. Does it make sense to get the $\{i,j\}$ entry of $P(s)=e^{sQ}$? Or am I missing something? Am I supposed to use the Chapman-Kolmogorov equations for this, if so: how?

Answer 1

Does it make sense to get the $\{i,j\}$ entry of $P(s)=e^{sQ}$?

Yes it does, and this is the definition, actually.