Definition (Q-matrix)

Let $I$ be a state space. A matrix $Q=(q_{i,j}:i,j\in I)$ is said to be a $Q$-matrix if

1. $q_{ij}\ge0$ for $i\not=j$
2. $-q_{ii}=\sum\limits_{j\not=i}q_{ij}<\infty$ for $i\in I$

Denote $\nu_{i}=-q_{ii}$ which can be interpreted as the rate of leaving $i$. We also call $q_{ij}$ the rate of transiting from $i$ to $j$.

Definition (Transition Matrix for Q)

Let $Q$ be a Q-Matrix. Define the associated transition matrix $\Pi=(\Pi_{ij}:i,j\in I)$ for the jump chain as follows:

1. If $\nu_{i}\not=0$, then $$\Pi_{ij}=\begin{cases}\frac{Q_{ij}}{\nu_{i}}&j\not=i\\0&j=i\end{cases}$$
2. If $\nu_{i}=0$, then $$\Pi_{ij}=\begin{cases}0&j\not=i\\1&j=i\end{cases}$$