Quadratic covariation and an extension of Itô's formula

Abstract

Let $X$ be a standard Brownian motion. We show that for any locally square integrable function $f$ the quadratic covariation $\left[f\right(X),X]$ exists as the usual limit of sums converging in probability. For an absolutely continuous function $F$ with derivative $f$, Itô's formula takes the form $F\left(X_t\right)=F\left(X_0\right)+{\int }_0^{t}f\left(X_s\right)d{X}_s+\frac{1}{2}\left[f\right(X),X{]}_t$. This is extended to the time-dependent case. As an example, we introduce the local time of Brownian motion at a continuous curve.