Chapman Kolmogorov Th. - generality?

1. Nov 23, 2015

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:

2. Nov 28, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 29, 2015

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