Riemann-Stieltjes approximations of stochastic integrals

Abstract

We consider the space C[0, 1] together with its Borel σ-algebra A and a Wiener measure P. Let Ω denote a point in C[0, 1] and let x(Ω, t) denote the coordinate process. Then, {x(Ω, t), tε[0, 1]} is a Wiener process, and stochastic integrals of the form {Mathematical expression} can be defined for a suitable class of φ{symbol}. In this paper we consider a sequence of Stieltjes integrals of the form {Mathematical expression} where {Ωn(Ω)} is a sequence of polygonal approximations to co. Conditions are found which ensure the quadratic-mean convergence of {In}, and the limit is expressed as the sum of the stochastic integral {Mathematical expression} and a "correction term". © 1969 Springer-Verlag.