Chapman-Kolmogorov equation

In mathematics, specifically in probability theory, and yet more specifically in the theory of Markovian stochastic processes, the Chapman-Kolmogorov equations can be viewed as an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.

These equations are pivotal in the study of this field, and they were worked out independently by the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov. To give some idea of their importance, they are just as important, or more so, than the Cauchy-Riemann equations in the subject of complex variables.

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


 * $$p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)$$

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


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

i.e. a straightforward marginalization over the nuisance variable.

(Note that we have not yet assumed anything about the temporal (or any other) ordering of the random variables -- the above equation applies equally to the marginalization of any of them).

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 $$i_1<\ldotsj$$. So, the Chapman-Kolmogorov equation takes the form
 * $$p_{i_3;i_1}(f_3\mid f_1)=\int_{-\infty}^\infty p_{i_3;i_2}(f_3\mid f_2)p_{i_2;i_1}(f_2\mid f_1)df_2.$$

When the probability distribution on the state space of a Markov chain is discrete and the Markov chain is homogeneous, 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 Xt is the state of the process at time t, then for any two points i and j in the state space, we have


 * $$P_{ij}(t)=P(X_t=j\mid X_0=i).$$