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. In the Markov chain setting, one assumes that . Then, because of the Markov property,

where the conditional probability is the transition probability between the times i > j). So, the Chapman-Kolmogorov equation takes the form

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:

P(t + s) = P(t)P(s)

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

See also

examples of Markov chains

External links

• The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
• Mathworld