Chapman–Kolmogorov Equation

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.

Suppose that { fi } 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 fn. Then, the Chapman–Kolmogorov equation is

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.)

Read more about Chapman–Kolmogorov EquationApplication To Time Dilated Markov Chains

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Chapman–Kolmogorov Equation - Application To Time Dilated Markov Chains
... process under consideration is Markovian, the Chapman–Kolmogorov equation is equivalent to an identity on transition densities ... So, the Chapman–Kolmogorov equation takes the form In English, and informally, this says that the probability of going from state 1 to state 3 can be found ... and the Markov chain is homogeneous, the Chapman–Kolmogorov equations can be expressed in terms of (possibly infinite-dimensional) matrix multiplication, thus where P(t) is the transition matrix of jump t, i.e ...

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