On Two Formulations of the Theory of Stochastic Processes Depending Upon a Continuous Parameter
J. L. Doob, Warren Ambrose · Annals of Mathematics · 1940
The theory of stochastic processes depending upon a continuous parameter is the theory of measure (probability) relations on a collection of functions, {x(t) }, t ranging over the real numbers. There is some difficulty however in finding an appropriate collection of functions on which to consider the measure relations. On the one hand it is desirable to consider as large a class of functions as possible while on the other hand it is desirable that the functions considered have enough regularity properties, both individually and as a class, that one may systematically make use of known theorems from the theory of functions in investigating these measure relations. One way of choosing the collection of functions to be considered has been given by Doob (II);' he considers first a measure defined on the space of all real-valued functions, x(t), and carries this measure over to certain subspaces. Then he shows that in certain cases these subspaces will have desirable regularity properties. Another approach was given by Wiener (V and VIII), who takes a function, f(t, x), subject to certain regularity conditions, and then considers the collection of t-functions obtained from f(t, x) by fixing x and allowing t to vary. Wiener defines a measure on this space of t-functions in terms of a measure on x-space. The principal result of the present paper is the establishing of some relations between these two approaches to the theory of stochastic processes depending upon a continuous parameter. In section 1 we give the precise formulations of these two kinds of stochastic processes, in section 2 we show their equivalence, and in section 3 we obtain some further theorems relating the two.