Itô’s Lemma: The Taylor Expansion for Stochastic Calculus

Students often neglect the

\frac{1}{2} \sigma^2 f_{xx} dt

term, treating

dW_t^2

as zero. Remember: Brownian paths have infinite variation, meaning higher-order terms like

(dW_t)^2

are not negligible; they provide the essential 'volatility adjustment' that defines stochastic calculus.

Institutional Reference: Advanced Stochastic Processes