The Itô Formula

In ordinary integration theory, we rarely need the explicit definition of an integral as a limit of Riemann sums. Rules such as the fundamental theorem of calculus and the substitution rule make ordinary integrals much easier to handle. The Itô integral would be of limited practical interest if no similar calculus were available. The central result is the Itô formula, which can be viewed as the stochastic analogue of the fundamental theorem of calculus.

The second integral term is the essential difference between Itô calculus and ordinary calculus.

The formula can be interpreted as a decomposition of $f(B_t)$ into a trend component and directionless noise. The first integral is an Itô integral with an $L^2_{\mathrm{loc}}$ integrand and is therefore a local martingale. The second integral has bounded total variation locally and represents the trend.