Forward Kolmogorov Equations

In a continuous time Markov chain {X(t)}, define p _ij (t) as the probability that X(t + s) = j, given that X(s) = i, for s, t ≥ 0, and r _ij as the transition rate out of state i to state j. Then Kolmogorov’s forward equations say that, for all states i,j and times t ≥ 0, dp _ij (t)/dt = $\sum$_k≠jr _kj p _ik (t) − $\nu$ _j p _ij (t), where $\nu$ _k is the transition rate out of state k, $\nu$ _k = $\sum$ _j r _kj .

See

Markov Chains

Markov Processes