Chapman-Kolmogorov equation

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In mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the Chapman-Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.

Suppose that {f_i} is an indexed collection of random variables, that is, a stochastic process. Let

be the joint probability density function of the values of the random variables f₁ to f_n. Then, the Chapman-Kolmogorov equation is

Particularization to Markov Chains

When the stochastic process under consideration is Markovian, the Chapman-Kolmogorov equation is equivalent to an identity on transition densities.

When the probability distribution on the state space of a Markov chain is discrete, the Chapman-Kolmogorov equations can be expressed in terms of (possibly infinite-dimensional) matrix multiplication, thus:

where P(t) is the transition matrix, i.e., if X_t is the state of the process at time t, then for any two points i and j in the state space, we have