Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators
type: Item
instance of: scholarly article
title: Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators
zbMATH Open document ID: 0955.60074
DOI: 10.2307/3318581
author: Gonzalo Perera
author: José R. León
published in: Bernoulli
publication date: 2001-02-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1081788033
review text: The trajectories of a particle of mass \(1/\beta\) submitted to the action of a harmonic oscillator with an external white noise force converge, as \(\beta\to\infty\), to an Ornstein-Uhlenbeck process [see \textit{E. Nelson}, ``Dynamical theories of Brownian motion'' (1967; Zbl 0165.58502)]. The authors of the present paper show that the number of crossings of the particle with a fixed level, suitably normalized, converge in \(L^2\) to the local time of the Ornstein-Uhlenbeck process. They also obtain a weak limit theorem indicating the speed of the convergence. A key to these results is the interpretation of the approximating process as a regularization of the Ornstein-Uhlenbeck process by convolution with a non-compactly supported kernel.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1455542
zbMATH Keywords: harmonic oscillator
zbMATH Keywords: external white noise force
zbMATH Keywords: Ornstein-Uhlenbeck process
zbMATH Keywords: local time
zbMATH Keywords: estimator
MaRDI profile type: Publication
OpenAlex ID: W1998313236
description: scientific article; zbMATH DE number 1455542