Stochastic integral

Abstract

Comm. by S. K.KEYA, M.I.A., Oct. 12, 1944.) 1. 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. 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 <... 1-.At first for f(t, )eS we define the stochastic integral (r, ,o) ,o) (for brevity denote it by I(t, ;f)) as follows" d,g(r,The cost of this research has been defrayed from the Scientific Expenditure of the Department of Education.1) C.P. Lvy" Thorie de l'addition des variable alatoire, P. 167, 1937, and also J. L. Doob" Stochastic processes depending on a continuous parameter, Trans., &mer.Math.Soc.vol.42, Theorem 3.9.2) , denotes the mathematical expectation, viz.'f()=I # f o)P(d).3) R.

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