Generalization of Ito's formula for smooth nondegenerate martingales

In this paper we prove the existence of the quadratic covariation [(partial derivativeF/partial derivativex(k))(X), X-k] for all 1 less than or equal to k less than or equal to d, where F belongs locally to the Sobolev space W-1,W-p(R-d) for some p > d and X is a d-dimensional smooth nondegenerate m artingale 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 ex tension of Ito's formula where the complementary term is one-half the sum o f the quadratic covariations above. (C) 2001 Elsevier Science B.V. All righ ts reserved. MSG: 60H05; 60H07.