Generalization of Itôʹs formula for smooth nondegenerate martingales

In this paper we prove the existence of the quadratic covariation [(∂F/∂xk)(X), Xk] for all 1⩽k⩽d, where F belongs locally to the Sobolev space W1,p(Rd) 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 Itôʹs formula where the complementary term is one-half the sum of the quadratic covariations above.