Representations and classifications of stochastic processes

Johnson, Dudley Paul

doi:10.1090/S0002-9947-1974-0331490-X

Representations and classifications of stochastic processes
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Trans. Amer. Math. Soc. 188 (1974), 179-197 Request permission

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