For discrete time finite Markov chain, the first passage time $T_j$ to visit state $j$, is determined from the recurrence equation: $ p^{(n)}_{ij} = \sum_{k=0}^n f_{ij}^{(k)} p^{(n-k)}_{jj} = \sum_{k=0}^n \mathbb{P}(T_j = k ; X_0 = i) \mathbb{P}(X_{n-k} = j ; X_0 = j) $ where $p^{(n)}_{ij} = \mathbb{P}(X_n = j ; X_0 = i)$ and $f^{(k)}_{ij} = \mathbb{P}(T_j = k ; X_0 = i)$. This recurrence equation allows to find probability generating function for the first passage time distribution (exerices 1.5.3 of J.R. Norris's book on "Markov Chains", relevant chapter 1.5 is available from Norris's website).
It strikes me odd that no book I have seen discusses first passage time distribution for continuous time finite Markov chain.
Can anyone suggest a reference where this is discussed/worked-out, or hint me at how to do this for myself.
I suspect a similar integral equation would be set-up, and moment generating function for the first-passage-time distribution would satisfy some simple equation.
Thanks for reading.