What is Generator matrix, jump chain?

The generator matrix of a continuous-time chain is $\mathcal L = (q_{ij})$ with $q_{ij} = \lim_{t\downarrow0}P_{ij}(t)/t \ge 0$ for $j \ne i$, the rate of jumps from $i$ to $j$, and $q_{ii} = -\sum_{j\ne i}q_{ij}$. The jump chain is the discrete-time chain of the states visited, with transition probabilities $q_{ij}/q_i$, where $q_i = -q_{ii}$.