Kolmogorov backward equation: what is it computing?

Question

The kolmogorov backward equation states:

$P_{ij}^{'}(t) = \sum_{k \ne i} q_{ik}P_{kj}(t) - v_iP_{ij}(t)$

Is this computing the rate of transition from i to j?

Answer 1

Actually, no. With the rates of transition $q_{i,j}$ given, this system of equations determines the transition probabilities $P_{i,j}(t)$.