 Solution

A Q-matrix satisfies $q_{ij}\ge 0$ for $i\ne j$ and $q_{ii}=-{\sum }_{j\ne i}{q}_{ij}$; on a countable state space the exit rate $-q_{ii}$ is required to be finite. For a finite-state continuous-time Markov chain with transition semigroup $P(t)$,The backward and forward Kolmogorov equations are respectively