# Riemann-Stieltjes approximations of stochastic integrals

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

1969

AUTHORS 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} can be defined for a suitable class of ϕ. In this paper we consider a sequence of Stieltjes integrals of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$I_n = \int\limits_0^1 \varphi {\text{ }}(\omega ^n (\omega ),t)dx(\omega ^n (\omega ),t)$$ \end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\int\limits_0^1 \varphi {\text{ }}(\omega ,t)dx(\omega ,t)$$ \end{document} and a “correction term”. More... »

PAGES

87-97

### Journal

TITLE

Probability Theory and Related Fields

ISSUE

2

VOLUME

12

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf00531642

DOI

http://dx.doi.org/10.1007/bf00531642

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1007146725

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