DES - Demonstration of the Chapman-Kolmogorov Equation

Arguments

• Definition of $p_{i,j}(t)$
• DES - Theorem ‘Total Probability Rule’
• DES - Definition of ‘Markov Process’
• DES - Chapman-Kolmogorov Equation

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Summary

Starting from:

$$p_{i,j}\triangleq P(X(s+t)=j\mid X(s)=i)$$

We want to arrive at:

$$p_{i,j}=\sum _{r\in \mathcal{X}}P(X(s+t)=j\mid X(s+u)=r)\cdot P(X(s+u)=r\mid X(s)=i)$$

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In matrix form this is equal to say:

$$H(t)=H(u)\cdot H(t-u)$$