Une remarque sur l'approximation de l'integrale stochastique du type noncausal par une suite des integrales de Stieltjes
type: Item
instance of: scholarly article
title: Une remarque sur l'approximation de l'integrale stochastique du type noncausal par une suite des integrales de Stieltjes
zbMATH Open document ID: 0551.60058
author: Shigeyoshi Ogawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: If \(f(t,\omega)\), (t\(\in [0,1]\), \(\omega\in \Omega ({\mathcal F},P))\), is a measurable process with \(P\{\int^{1}_{0}f^ 2(t,\omega)dt<\infty \}=1\), \(B(t,\omega)\) is a Brownian motion and \(\{\phi_ n\}\) is an orthonormal basis of \(L^ 2([0,1])\), then the series \(\sum <f,\phi_ n><\phi_ n,\dot B>\) with convergence in probability is used to define a stochastic integral \(\int^{1}_{0}f(t,\omega)d^*B(t,\omega)\). While this definition avoids the customary nonanticipatory condition, it may depend on the choice of the base. The author addresses to this difficulty and proves the following result: If the above process f is integrable in the \(L^ 1(\Omega)\) sense w.r.t. a trigonometric system in \(L^ 2[0,1]\), then it is also integrable w.r.t. the Haar system, and the two integrals are equal.
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 3878097
zbMATH Keywords: approximation
zbMATH Keywords: Stieltjes integrals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228902
OpenAlex ID: W1981674971
cites work: Q4181047
DOI: 10.2748/TMJ/1178228902
description: scientific article; zbMATH DE number 3878097