Let $X$ be a continuous time Markov process with finite state space $S$ and let $\mu_{ij}(t)$ be the transition rates. Denote by $I_i(t)={\bf 1}_{\{X_t=i\}}$ and $N_{ij}(t)=\#\{s\in [0,t]: X_{s-}=i, X_s=j\}$ where $\#$ is the counting measure, i.e. $N_{ij}(t)$ counts the number of jumps from $i$ to $j$ on $[0,t]$.
I would like to know if there is a way to compute the following conditional expectations $$\mathbb{E}[N_{ij}(s)|N_{kl}^t],\quad \mathbb{E}[I_i(s)|N_{jk}^t], \quad i,j,k\in S,$$ where $N_{ij}^t$ denotes the stochastic process $N_{ij}$ stopped at time $t$, i.e. $N_{ij}^t(s)=N_{ij}(s){\bf 1}_{\{s\leq t\}}+N_{ij}(t){\bf 1}_{\{s> t\}}$.
I learnt how to do it if $S=\{0,1\}$ and $1$ is absorbing, since then $I=1-N$, but is there any trick to do this computation in general?