Suppose $X_t$ is a continuous time, time homogeneous, ergodic Markov chain taking values in the state space $\{x,y\}$. Suppose $X_t$ is characterized by the forward equation $$ \frac{dp_t}{dt}=Ap_t , $$ where $p_t$ is the transition probability of $X_t$ at time $t$.
- Is there a closed-form solution for the first passage (hitting) time to go from state $x$ to state $y$? I've seen a few solutions to this problem in discrete time but I wasn't sure how to translate them to continuous time.
- More interestingly, if $q \in [0,1]$ then how can I calculate the hitting time $$ \inf \left\{\Delta \geq 0 : \mathbb{P}\left( \cup_{\delta \in [0,\Delta]} \{ X_t=X_{t+\delta} \} |X_t \right)\geq q \right\}? $$