On Itô's formula for multidimensional Brownian motion
type: Item
instance of: scholarly article
title: On Itô's formula for multidimensional Brownian motion
zbMATH Open document ID: 0955.60077
DOI: 10.1007/PL00008719
author: Hans Föllmer
author: Philip Protter
publication date: 2000-06-07T00:00:00Z
review text: This paper is a generalization in greater dimensions of a result already due to the authors together with \textit{A. N. Shiryayev} [Bernoulli 1, No. 1/2, 149-169 (1995; Zbl 0851.60048)]. It deals with an extension of Itô's formula for Brownian motion: if \(F\) belongs locally to the Sobolev space \(W^{1,2}({\mathbb R}^d)\) and if \(X\) is a \(d\)-dimensional Brownian motion, then for any \(t\geq 0\) and any starting point \(x\in {\mathbb R}^d\) except in some polar set, \(F(X_t)\) decomposes into \[ F(X_t) = F(x) + \sum_{k=1}^d\int^t_0 f_k (X_s) dX_s+ \tfrac 12 \sum_{k=1}^d \bigl[ f_k (X), X^k \bigr]_t, \] where the \(f_k\) denote the (weak) partial derivatives of \(F\), and \(\bigl[ f_k(X), X^k \bigr]\) is a quadratic covariation term. Notice that in the above formula, \(F(X)\) may not be a semimartingale, so that the quadratic covariation term may not have bounded variations as in the classical Itô's formula. However \(F(X)\) is a Dirichlet process, and this quadratic covariation term is indeed the process of zero energy appearing in \textit{M. Fukushima}'s decomposition [``Dirichlet forms and Markov processes'' (1980; Zbl 0422.31007)]. It should also be mentioned that the condition on the starting point is not at all a restriction, since it is also needed to define the stochastic integrals along \(X\) of the \(f_k (X)\). Regarding the proof, the main argument consists in establishing that the quadratic variation term indeed exists, if \(F\) belongs locally to \(W^{1,2}({\mathbb R}^d)\). This is done by using an approximation in terms of backward and forward stochastic integrals (which leads also, finally, to a change of variable formula of Stratonovich type), and a multidimensional analogue of the 0-1 law of Engelbert and Schmidt [see the paper of \textit{R. Höhnle} and \textit{K.-Th. Sturm}, Stochastics Stochastics Rep. 44, No. 1/2, 27-41 (1993; Zbl 0780.60078)].
reviewed by: Thomas Simon
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 1406471
zbMATH Keywords: Itô's formula
zbMATH Keywords: Brownian motion
zbMATH Keywords: stochastic integrals
zbMATH Keywords: quadratic covariation
zbMATH Keywords: Dirichlet spaces
zbMATH Keywords: polar sets
MaRDI profile type: Publication
full work available at URL: pl00008719
OpenAlex ID: W2090566355
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1406471