Transformation de Fourier et temps d'occupation browniens. (Fourier transformation and Brownian occupation time)
type: Item
instance of: scholarly article
title: Transformation de Fourier et temps d'occupation browniens. (Fourier transformation and Brownian occupation time)
zbMATH Open document ID: 0717.60092
DOI: 10.1007/BF01212557
publication date: 1991-01-01T00:00:00Z
review text: We study oscillatory stochastic integrals of the form \[ \Gamma (\lambda)=\int^{\infty}_{0}\exp (i\lambda B_ s)g(s)ds, \] where \(\lambda\) is a nonzero parameter and g a square integrable function. We study integrability properties of \(\Gamma\) (\(\lambda\)) and its behavior as a function of \(\lambda\), using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey... We also obtain limit theorems in law related to the variables \(\Gamma\) (\(\lambda\)) based upon an asymptotic version of a theorem of \textit{F. B. Knight} [A reduction of continuous square integrable martingales to Brownian motion, in: H. Dinges (ed.), Martingales. A report on a meeting at Oberwolfach, May 17- 23, 1970 (1971; Zbl 0226.60070)] on orthogonal continuous martingales. We consider the random measure, image by the Brownian motion of the unbounded measure \(1_{[0,\infty]}(s)g(s)ds\); we prove the existence and the continuity of an occupation time density. Finally, under a stronger integrability condition on g, we show the existence of a density for the law of \(\Gamma\) (\(\lambda\)), using Malliavin's calculus.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 4182553
zbMATH Keywords: oscillatory stochastic integrals
zbMATH Keywords: integrability properties
zbMATH Keywords: limit theorems in law
zbMATH Keywords: occupation time density
zbMATH Keywords: Malliavin's calculus
author: Catherine Donati-Martin
MaRDI profile type: Publication
cites work: Continuity of local times for L�vy processes
cites work: Le problème de Buffon–Synge pour une corde
cites work: Q3241504
cites work: On tail probabilities for martingales
cites work: Q5665949
cites work: Semi-martingales et grossissement d'une filtration
cites work: On limit processes for a class of additive functional of recurrent diffusion processes
cites work: Q3334701
cites work: Étude asymptotique de certains mouvements browniens complexes avec drift
cites work: Stochastic calculus with anticipating integrands
cites work: Q4169502
cites work: Asymptotic laws of planar Brownian motion
cites work: Q3862204
full work available at URL: https://doi.org/10.1007/bf01212557
OpenAlex ID: W2912596261
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article