 Kolmogorov equations

For a finite-state continuous-time Markov chain with Q-matrix $Q$, the backward and forward equations are $P^{\prime }(t)=QP(t)$ and $P^{\prime }(t)=P(t)Q$, respectively.

Kolmogorov equations refer primarily to a set of differential equations that describe the evolution of probabilities in stochastic processes, particularly in the contexts of Markov processes and stochastic differential equations. These equations are pivotal in the study of probability theory and were developed by the Russian mathematician Andrey Kolmogorov.