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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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