I need help with this problem about return times in continuous time Markov chains:
We are given a continuous time Markov chain $(X_t)$. Define $T_i=\inf{\{t>0; X_t=i, X_{t-} \neq i\}}$, which is the first return time to $i$. Define $N_j(t)$ to be the number of visits to state $j$ in time $t$.
Want to prove: $$E\left[N_j(T_i)\mid X_0 = i\right] = \frac{P(T_j < T_i \mid X_0=i)}{P(T_i < T_j \mid X_0=j)}$$
So, because people are complaining, I'm going to write how I tried to solve it:
Yes, this is a problem I found interesting because of the symmetry involved in the fraction on the right. My first attempt to show this was to write each of the $P(T_j < T_i \mid X_0=i)$ as a sum of $\pi_{jk} P(T_j < T_i \mid X_0=k)$ terms, where $\pi_{jk}$ denote the jump probabilities, but it seems to complicate the matter more than illuminate it. Then I tried to gain something from the connection between the continuous and discrete notions of "time spent in $j$", as they differ by a factor $q_j$, which is the rate of staying in State $j$. Maybe starting with a case of 3 states would help?
I guess one needs to expand the LEFT hand side somehow...