Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0955.60077
Föllmer, Hans; Protter, Philip
On Itô's formula for multidimensional Brownian motion. (English)
[J] Probab. Theory Relat. Fields 116, No.1, 1-20 (2000). ISSN 0178-8051; ISSN 1432-2064/e

This paper is a generalization in greater dimensions of a result already due to the authors together with {\it 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 $W^{1,2}({\Bbb R}^d)$ and if $X$ is a $d$-dimensional Brownian motion, then for any $t\geq 0$ and any starting point $x\in {\Bbb R}^d$ 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 {\it 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 $W^{1,2}({\Bbb R}^d)$. 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 {\it R. Höhnle} and {\it K.-Th. Sturm}, Stochastics Stochastics Rep. 44, No. 1/2, 27-41 (1993; Zbl 0780.60078)].
[Thomas Simon (Berlin)]

MSC 2000:: *60J65 Brownian motion
31C25 Dirichlet spaces
60H05 Stochastic integrals
31C15 Generalizations of potentials, etc.

Keywords: Itô's formula; Brownian motion; stochastic integrals; quadratic covariation; Dirichlet spaces; polar sets

Citations: Zbl 0851.60048; Zbl 0422.31007; Zbl 0780.60078

Cited in: Zbl 0979.60071

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]