Stochastic Process

A stochastic process is a collection of random variables indexed by time. A stochastic process is completely specified by its finite joint distributions. The standard setup specifies a sample space \Omega, and a probability measure P defined on algebras \mathcal{A}_t, t\in T. It is common to assume T is totally ordered and \mathcal{A}_t\subseteq\mathcal{A}_u whenever t\le u.

We write X_t\colon\mathcal{A}_t\to\boldsymbol{R}, t\in T, to indicate X_t is a function from \Omega to the real numbers that is \mathcal{A}_t-measurable. If \mathcal{A} is an algebra of sets then A\in\mathcal{A} is an atom of \mathcal{A} if B\in\mathcal{A} and B\subseteq A then either B = A or B = \emptyset.

Exercise. If \mathcal{A} is a finite algebra of sets on \Omega then the atoms of \mathcal{A} are a partition of $.

Hint: A collection \mathcal{A} of subsets of \Omega is a partition if the sets in \mathcal{A} are pairwise disjoint and the union of all sets in \mathcal{A} equals \Omega. Given \omega\in\Omega let A_\omega = \cap\{A\in\mathcal{A}\mid \omega\in\mathcal{A}\}.

In the finite case the atoms of \mathcal{A} determine the algeba \mathcal{A} so we can identify \mathcal{A} with its atoms.

Recall that a function X\colon\Omega\to\boldsymbol{R} is \mathcal{A}-measurable if \{\omega\in\Omega\mid f(\omega)\le x\} belongs to \mathcal{A} for all x\in\boldsymbol{R}.

Exercise. If \mathcal{A} is finite show f is measurable if and only if it is constant on atoms of \mathcal{A}.

A consequence of this exercise is that f\colon\Omega\to\boldsymbol{R} is \mathcal{A}-measurable if and only if f\colon\mathcal{A}\to\boldsymbol{R} is a function from the atoms of \mathcal{A} to the real numbers.