I was reading a formulation of piecewise-deterministic Markov process $\Pi_t$, $ t \in \mathbb{R}_+$. In particular $\Pi_t$ is defined as $\Pi_t = P (X_t | \mathcal{F}_{\lfloor t/ \Delta \rfloor}) $, where $X_t$ is a continuous time homogeneous Markov Chain. I understand the part for $\Pi_{n \Delta} $, where $\Delta $ is some fixed time unit. Now I'm confused with the different formulation of for time $t \in (n \Delta, (n+1) \Delta)$ and the derivation is the following
\begin{equation} \frac{d\Pi_t}{dt} = lim_{h \rightarrow 0^+} \frac{\Pi_{t+h} - \Pi_t}{h} = lim_{h \rightarrow 0^+} \frac{E(I_{t+h} - I_t | \mathcal{F_{ \lfloor t/ \Delta \rfloor}})}{h} = \Pi_t Q \end{equation}
where the indicator function $I_t = [I_{\{X_t =1\}}, ..., I_{\{X_t $.= N+1\}}]$ , and $Q = p_{ij}$ is the matrix of transition probabilities of $X_t$ from state $ i $ to $j$.( Note that $X_t$ is continuous time homogenous, and for each $p_{ij} \in Q$,
$p_{ij} = lim_{h \rightarrow 0^+} \frac{P(X_{t+h} = j | X_t = i)}{h}$
So I don't understand how to derive the last equality , from the limit taking $h \rightarrow 0^+$ to $\Pi_t Q$ explicitly, although I conceptually feel the meaning of $\Pi_t Q $ indicates instaneous trasition of $X_t$ at time $t$. So can anyone help me see the derivation of the last equality ? Or does it simply follow from the meaning of the definition that Q generates the homogeneous Markov Chain $X_t$?
A second question is that from the derivation $\frac{d\Pi_t}{dt} = \Pi_t Q$, I know it can easily derive $\Pi_t = C \cdot exp(tQ)$. However it directly concludes $\Pi_t =\Pi_{\lfloor t/\Delta \rfloor \Delta} exp((t- \lfloor t/ \Delta \rfloor \Delta ) Q))$. I wasn't sure how to obtain this derivation with $\Pi_{\lfloor t / \Delta \rfloor \Delta}$, can anyone help with some insights ? Thanks very much!