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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: We show that to every stochastic process X one can associate a unique collection $ (\Phi ,{\Phi _ + },T(t),E(U),{p^\ast})$ consisting of a linear space $ \Phi $, on which is defined a linear functional $ {p^ \ast }$, together with a convex subset $ {\Phi _ + }$ which is invariant under the semigroup of operators $ T(t)$ and the resolution of the identity $ E(U)$. The joint distributions of X, there being one version for each $ \phi \in {\Phi _ + }$, are then given by

$\displaystyle {P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .$

To each $ \phi $ contained in the extreme points $ {\Phi _{ + + }}$ of $ {\Phi _ + }$ and each time t we find a probability measure $ P_t^ \ast (\phi, \cdot )$ on $ {\Phi _{ + + }}$ such that $ T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )$. $ P_t^ \ast $ is the transition probability function of a temporally homogeneous Markov process $ {X^ \ast }$ on $ {\Phi _{ + + }}$ for which there exists a function f such that $ X = f({X^ \ast })$. We show that in a certain sense $ {X^ \ast }$ is the smallest of all Markov processes Y for which there exists a function g with $ X = g(Y)$. 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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Keywords: Stochastic process, algebraic representation, Markov process representation, Choquet's theorem, dual process, $ \Xi $ process
Article copyright: © Copyright 1974 American Mathematical Society