Representations and classifications of stochastic processes

Author:
Dudley Paul Johnson

Journal:
Trans. Amer. Math. Soc. **188** (1974), 179-197

MSC:
Primary 60G05

MathSciNet review:
0331490

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that to every stochastic process *X* one can associate a unique collection consisting of a linear space , on which is defined a linear functional , together with a convex subset which is invariant under the semigroup of operators and the resolution of the identity . The joint distributions of *X*, there being one version for each , are then given by

*t*we find a probability measure on such that . is the transition probability function of a temporally homogeneous Markov process on for which there exists a function

*f*such that . We show that in a certain sense is the smallest of all Markov processes

*Y*for which there exists a function

*g*with . We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0331490-X

Keywords:
Stochastic process,
algebraic representation,
Markov process representation,
Choquet's theorem,
dual process,
process

Article copyright:
© Copyright 1974
American Mathematical Society