Itô’s Lemma in One Dimension

Why an extra term appears

Suppose $X_t$ follows

dX_t=\mu_t\,dt+\sigma_t\,dW_t,

and let $f(t,X_t)$ be smooth. In ordinary calculus, a first-order expansion would seem sufficient. Brownian motion changes the bookkeeping because $dW_t$ is of order $\sqrt{dt}$ , so $(dW_t)^2$ is of order $dt$ and cannot be discarded.

Keep the second order

A second-order Taylor expansion gives

df \approx f_t\,dt+f_x\,dX_t+\frac12 f_{xx}(dX_t)^2.

Substituting the dynamics of $X_t$ and retaining terms of order $dt$ leaves $(dX_t)^2\approx\sigma_t^2dt$ . The mixed and higher-order terms vanish in the limit. Therefore,

df= \left(f_t+\mu_t f_x+\frac12\sigma_t^2f_{xx}\right)dt +\sigma_t f_x\,dW_t.

The mental model

The stochastic chain rule is not ordinary calculus with random symbols pasted in. Its extra curvature term records the quadratic variation of Brownian paths. If $f$ is linear, $f_{xx}=0$ and the familiar first-order rule returns. Curvature is where the difference lives.