Kolmogorov equations (Markov jump process)

In the context of a continuous-time Markov process, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time-evolution of the probability ${\displaystyle P(x,s;y,t)}$, where ${\displaystyle x,y\in \Omega }$ (the state space) and ${\displaystyle t>s}$ are the final and initial time respectively.

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