Quadratic covariation and an extension of Itô's formula

Let X be a standard Brownian motion. We show that for any locally square integrable function f the quadratic covariation [ f(X),X] exists as the usual limit of sums converging in probability. For an absolutely continuous function F with derivative f , Ito's formula takes the form F (X t)=F(X 0)+∫ 0 tf(X s)dX s+1 2 [f(X),X] t . This is extended to the time-dependent case. As an example, we introduce the local time of Brownian motion at a continuous curve.