Stochastic integral

Introduction. Let (/2, P) be any probability field, and g (t, ), 0 ___< t 1, (oe/2, be any brownian motion 1) on (/2, P) i. e. a (real) stochastic differential process with no moving discontinuity such that (g (s, , o) -g (t, , o) ) =O) and (g (s, , , , ) -g (t, , o) ) =ls-tl. In this "note we shall investigate an integral _/. f (r, ) dg (r, o) for any element f (t, ) in a functional class S* which will be defined in 2; the particular case in which f (t, o) does not depend upon has already been treated by Paley and Wiener). In 2 we shall give the definition and prove fundamental properties concerning this integral. In 3 we shall establish three theorems which give sufficient conditions for integrability. In 4 we give an example, which will show a somewhat singular property of our integral. 2. Definition and Properties. For brevity we define the classes of measurable functions defined on 0, 1 /2" G, S (t0, tl,. . . , t), S and S* respectively as the classes of f (t, , o) satisfying the corresponding conditions, as follows, G" f (r, (), g (r, o), 0 r, are independent of g (a, o) -g (t, ), t a 1, for any t, g (r, o) being the above mentioned brownian motion, S (to, t, ,. . . , t, ), 0=t0 <: tt <.