Integration with respect to fractal functions and stochastic calculus. I
type: Item
instance of: scholarly article
title: Integration with respect to fractal functions and stochastic calculus. I
zbMATH Open document ID: 0918.60037
publication date: 1999-08-02T00:00:00Z
review text: The integral \(\int_a^b fdg\) of real or complex-valued functions of unbounded variation on a finite interval \((a,b)\) is defined using composition formulas and integration-by-parts rules for fractional integrals and Weyl derivatives. This integral is defined as \[ \int_a^b fdg=(-1)^\alpha \int_a^b D^\alpha_{a+} [f(x)-f(a+)]D^{1-\alpha}_{b-}[g(x)-g(b-)]dx+f(a+)(g(b-)-g(a+)), \] where \(f\) and \(g\) satisfy some fractional differentiability conditions in \(L_p\)-spaces, and \(\alpha\in [0,1]\). If \(g\) has bounded variation and \(f\) is bounded and right- (or left-)continuous at \(dg\)-almost all points (or satisfies a suitable integrability condition), then this integral coincides with the classical Lebesgue-Stieltjes integral. In the special case of Hölder continuous functions \(f\) and \(g\) of summed order greater than 1 then the convergence of the Riemann-Stieltjes sums to the integral is proved. Furthermore, the integral as a function of the boundary is Hölder continuous of the same order as \(g\). The paper contains some applications of the above results to stochastic calculus, obtained by replacing \(g\) by the trajectories of a Brownian motion \(W\). First it is proved that for adapted random \(L_2\)-functions \(f\) of fractional degree of differentiability greater than \(1/2\) the integral exists and coincides with the classical Itô integral with probability one. For the more general class of adapted functions having fractional derivatives of all orders less than 1/2, the author proves an approximation result for the Itô integral by the integrals of a regularization of \(f\). These results are extended to anticipating random functions \(f\) by means of the Wiener chaos expansion. In this case the integral defined in this paper is related with the forward integral, which can be expressed in terms of the Skorokhod integral and a complementary term involving the derivative operator [see \textit{D. Nualart} and \textit{E. Pardoux}, Probab. Theory Relat. Fiel…
reviewed by: David Nualart
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 26A42
zbMATH DE Number: 1203669
zbMATH Keywords: fractional integrals
zbMATH Keywords: fractional derivatives
zbMATH Keywords: stochastic integrals
author: Martina Zähle
MaRDI profile type: Publication
full work available at URL: s004400050171
OpenAlex ID: W2003427294
DOI: 10.1007/S004400050171
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1203669