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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 $$W^{1,2}({\mathbb R}^d)$$ 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 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 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 forms 60H05 Stochastic integrals 31C15 Potentials and capacities on other spaces
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