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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 ( d ) and if X is a d-dimensional Brownian motion, then for any t0 and any starting point x d except in some polar set, F(X t ) decomposes into

F(X t )=F(x)+ k=1 d 0 t f k (X s )dX s +1 2 k=1 d f k (X) , X k t ,

where the f k denote the (weak) partial derivatives of F, and f k (X) , X k 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 ( 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:
60J65Brownian motion
31C25Dirichlet spaces
60H05Stochastic integrals
31C15Generalizations of potentials and capacities