Itô's lemma
Itô's lemma is the stochastic analogue of the chain rule. If
\[dX_t=a\,dt+b\,dW_t\]and \(Y_t=f(X_t,t)\), then
\[df=\left(f_t+a f_x+\frac12b^2f_{xx}\right)dt+b f_x\,dW_t.\]The extra second-derivative term appears because Brownian increments satisfy the quadratic-variation rule \((dW_t)^2=dt\) in Itô calculus.
Key idea. Ordinary chain-rule differentiation is incomplete for functions of stochastic processes; Itô's lemma adds the required correction.