Generalization of Itô's formula for smooth nondegenerate martingales

Abstract

In this paper we prove the existence of the quadratic covariation $\text{[(∂F/∂x}{}_{\mathrm{k}}\text{)(X),}\text{X}{}^{\mathrm{k}}\text{]}$ for all 1⩽k⩽d, where F belongs locally to the Sobolev space $\text{W}{}^{\mathrm{1,p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ 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.