On Ito's formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X-1 , . . . , X-d) and a func tion F which belongs locally to the Sobolev space W-1,W-2. We prove an exte nsion of Ito's formula where the usual second order terms are replaced by t he quadratic covariations [f(k) (X), X-k] involving the weak first partial derivatives f(k) of F. In particular we show that for any locally square-in tegrable function f the quadratic covariations [f (X), X-k] exist as limits in probability for any starting point, except for some polar set. The proo f is based on new approximation results for forward and backward stochastic integrals.