Itô's lemma

Remarks

The proof expands $f$ by Taylor's formula to second order over a fine partition of $[0, t]$. The sums of $(\Delta X)^2$ converge to $\int \sigma_s^2 \, ds$, the quadratic variation of $X$, while higher-order terms vanish in the limit. In differential form, $df = \partial_t f \, dt + \partial_x f \, dX + \tfrac12 \partial_{xx} f \, (dX)^2$ with the rules $(dB)^2 = dt$ and $dt \, dB = (dt)^2 = 0$. Example: for $f(x) = x^2$ and $X = B$ with $B_0 = 0$, it gives $B_t^2 = 2 \int_0^t B_s \, dB_s + t$. Example: for geometric Brownian motion $dS = \mu S \, dt + \sigma S \, dB$ with constants $\mu, \sigma$ and $S_0 > 0$, the function $\log$ gives $\log S_t = \log S_0 + (\mu - \tfrac12 \sigma^2) t + \sigma B_t$. The $C^2$ hypothesis matters: for $f(x) = \abs{x}$ the formula needs an extra local time term, which is Tanaka's formula.

Going deeper

For a student

The lemma is the main computational tool of stochastic calculus. It solves linear stochastic differential equations, such as those for geometric Brownian motion and the Ornstein–Uhlenbeck process. It links expectations of diffusions to partial differential equations, as in the Kolmogorov backward equation and the Feynman–Kac formula. The multidimensional and semimartingale versions have the same shape, with the quadratic covariations $d[X^i, X^j]$ in the second-order term.