Itô's Lemma: The Extra Second-Order Drift

In ordinary calculus, the differential of f of x is f prime of x times d x. But when x is a Brownian motion, its quadratic variation is non zero, and ordinary calculus fails.

Ito's lemma expands f to second order. Because d W squared equals d t, the second derivative term does not vanish. It adds a deterministic drift of half f double prime d t.

The rest of the narration is in the book's Companion.