An extension of Stieltjes-Young integrals

Let ƒﾓ(t) and f(t) be real-valued functions defined on a closed interval [a, b]. The Riemann-Stieltjes integral of f with respect to ƒﾓ is usually denoted by [numerical formula] When ƒﾓ(x) is of bounded variation on the interval [a, b], we can treat this integral in the framework of measure theory. Let p and q are positive numbers such that [numerical formula] L. C. Young showed that the integral [numerical formula] in the case that f(t) and ƒﾓ(t) have finite mean variation of order p and q, respectively. In this paper we shall try to extend the Stieltjes-Young integration theory when f(t) and ƒﾓ(t) are stochastic processes.