The Chapman Kolmogorov Relation

The Chapman Kolmogorov Relation

The Chapman Kolmogorov relation is an important result in the theory of (discrete) Markov chains as it provides a method for calculating the $n$-step transition probability matrix of a Markov chain from the 1-step transition probability matrix of a Markov chain. The Chapman-Kolmogorov relation can be written as follows: $$ \mathbf{P}^{n+m} = \mathbf{P}^n\mathbf{P}^m $$ Here $\mathbf{P}^{n+m}$ is the $n+m$-step transition probability matrix, $\mathbf{P}^{n}$ is the $n$-step transition probability matrix and $\mathbf{P}^{m}$ is the $m$-step transition probability matrix. The above equation holds for discrete Markov chains. For continuous time Markov chains the elements of the transition probability matrix are written as a function of time. For continuous time Markov chains the Chapman Komogorov relation is written as: $$ \mathbf{P}(t + s ) = \mathbf{P}(t)\mathbf{P}(s) $$

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