Chapman Kolmogorov Th. - generality?

[Stephen Tashi]

Does the Chapman - Kolmogorov equation hold for an arbitrary stochastic process?

The current wikipedia article on "Stochastic process" ( https://en.wikipedia.org/wiki/Stochastic_process ) seems to say that the Chapman-Kolmogorov equation is not valid for an arbitrary stochastic process.

I say "seems to say" since I can't interpret what the article means by "same class" in the passage:

Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the full-blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a Wiener process (in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. When this condition is expressed in terms of probability densities, the result is called the Chapman–Kolmogorov equation.

 

[Stephen Tashi]

Whether the equation is true for an arbitrary process depends on what one calls "the Chapman-Kolmogorov Equation". In "Handbook Of Stochastic Methods" by C.W. Gardiner, second edition, pages 43-44, marginalization is a proof for the equation:

$$p(x_1,t_1) = \int dx_2 \ p(x_1,t_1; x_2, t_2)$$

which corresponds to the equation in the current Wikipedia article given by:

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

which that article calls "the Chapman-Kolmogorov equation".

However, Gardiner does not call that equation "the Chapman-Kolmogorov Equation".

Marginalization also proves the equation

$$p(x_1,t_1\ |\ x_3,t_3) = \int dx_2\ p(x_1,t_1; x_2,t_2|x_3,t_3)$$
$$= \int dx_2\ p(x_1,t_1| x_2,t_2; x_3,t_3)\ p(x_2,t_2 | x_3,t_3)$$

Gardiner says

This equation is also always valid. We now introduce the Markov assumption. If $t_1 \ge t_2 \ge t_3$ we can drop the $t_3$ dependence in the double conditioned probability and write

$$p(x_1,t_1 | x_3,t_3) = \int dx_2\ p(x_1,t_1| x_2,t_2) p(x_2,t_2| x_3,t_3)$$

which is the Chapman-Kolmogorov equation.