Statement

If $f\in C^2$, then

$$ f(B_T)-f(B_0)

\int_0^T f’(B_t),dB_t + \frac12\int_0^T f’’(B_t),dt. $$

Differential form

$$ df(B_t)=f'(B_t)\,dB_t+\frac12 f''(B_t)\,dt. $$

Interpretation

Compared with the ordinary chain rule, Brownian motion introduces an extra correction term involving $f''$.