The intrinsic local time sheet of Brownian motion
type: Item
instance of: scholarly article
title: The intrinsic local time sheet of Brownian motion
zbMATH Open document ID: 0722.60079
DOI: 10.1007/BF01418866
author: L. C. G. Rogers
author: John B. Walsh
publication date: 1991-01-01T00:00:00Z
review text: McGill showed that the intrinsic local time process \(\tilde L(t,x)\), \(t\geq 0\), \(x\in {\mathbb{R}}\), of one-dimensional Brownian motion is, for fixed \(t>0\), a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4190866
zbMATH Keywords: local time process
zbMATH Keywords: Doob-Meyer decomposition
zbMATH Keywords: Brownian local time sheet
zbMATH Keywords: Dirichlet process
zbMATH Keywords: semimartingale
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01418866
OpenAlex ID: W2078771324
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4190866