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

<math>p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)<math>

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

<math>p_{i_1,\ldots,i_{n-1}}(f_1,\ldots,f_{n-1})=\int_{-\infty}^{\infty}p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)\,df_n<math>

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 <math>i_1<\ldotsMarkov property,

<math>p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)=p_{i_1}(f_1)p_{i_2;i_1}(f_2\mid f_1)\cdots p_{i_n;i_{n-1}}(f_n\mid

f_{n-1}),<math> where the conditional probability <math>p_{i;j}(f_i\mid f_j)<math> is the transition probability between the times <math>i>j<math>). So, the Chapman-Kolmogorov equation takes the form

<math>p_{i_3;i_1}(f_3\mid f_1)=\int_{-\infty}^\infty p_{i_3;i_2}(f_3\mid f_2)p_{i_2;i_1}(f_2\mid f_1)df_2.<math>

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:

<math>P(t+s)=P(t)P(s)<math>

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

<math>P_{ij}(t)=P(X_t=j\mid X_0=i).<math>

See also

examples of Markov chains

External links

• The Legacy of Andrei Nikolaevich Kolmogorov (http://www.kolmogorov.com/) 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 (http://mathworld.wolfram.com/Chapman-KolmogorovEquation.html)