Abstract
In this paper we prove the existence of the quadratic covariation [(∂F/∂x k)(X), X k] for all 1⩽ k⩽ d, where F belongs locally to the Sobolev space W 1,p( R d) for some p> d and X is a d-dimensional smooth nondegenerate martingale adapted to a d-dimensional Brownian motion. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. As a consequence we obtain an extension of Itô's formula where the complementary term is one-half the sum of the quadratic covariations above.
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