Chapman–Kolmogorov equation

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In mathematics, specifically in probability theory and in particular the theory of Markovian stochastic processes, the Chapman–Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was arrived at independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov.

Mathematical description[edit]

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.)

The differential form of the Chapman–Kolmogorov equation is known as master equation.

Application to Time Dilated Markov chains[edit]

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 i1 < ... < in. Then, because of the Markov property,

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}),

where the conditional probability p_{i;j}(f_i\mid f_j) is the transition probability between the times i>j. 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.

Informally, this says that the probability of going from state 1 to state 3 can be found from the probabilities of going from 1 to an intermediate state 2 and then from 2 to 3, by adding up over all the possible intermediate states 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 of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps.

As a corollary, it follows that to calculate the transition matrix of jump t, it is sufficient to raise the transition matrix of jump one to the power of t, that is

P(t)=P^t.\,

See also[edit]

References[edit]