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On Itô’s formula for multidimensional Brownian motion. (English) Zbl 0955.60077

This paper is a generalization in greater dimensions of a result already due to the authors together with 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 ${W}^{1,2}\left({ℝ}^{d}\right)$ and if $X$ is a $d$-dimensional Brownian motion, then for any $t\ge 0$ and any starting point $x\in {ℝ}^{d}$ except in some polar set, $F\left({X}_{t}\right)$ decomposes into

$F\left({X}_{t}\right)=F\left(x\right)+\sum _{k=1}^{d}{\int }_{0}^{t}{f}_{k}\left({X}_{s}\right)d{X}_{s}+\frac{1}{2}\sum _{k=1}^{d}{\left[{f}_{k}\left(X\right),{X}^{k}\right]}_{t},$

where the ${f}_{k}$ denote the (weak) partial derivatives of $F$, and $\left[{f}_{k}\left(X\right),{X}^{k}\right]$ is a quadratic covariation term. Notice that in the above formula, $F\left(X\right)$ may not be a semimartingale, so that the quadratic covariation term may not have bounded variations as in the classical Itô’s formula. However $F\left(X\right)$ is a Dirichlet process, and this quadratic covariation term is indeed the process of zero energy appearing in 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 ${f}_{k}\left(X\right)$. Regarding the proof, the main argument consists in establishing that the quadratic variation term indeed exists, if $F$ belongs locally to ${W}^{1,2}\left({ℝ}^{d}\right)$. 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 R. Höhnle and K.-Th. Sturm, Stochastics Stochastics Rep. 44, No. 1/2, 27-41 (1993; Zbl 0780.60078)].

##### MSC:
 60J65 Brownian motion 31C25 Dirichlet spaces 60H05 Stochastic integrals 31C15 Generalizations of potentials and capacities