Proof of Itô's Formula

Step 1: Taylor expand $f(X_{t+h}) - f(X_t)$ to second order:

$f(X_{t+h}) - f(X_t) \approx f'(X_t)(X_{t+h} - X_t) + \frac{1}{2}f''(X_t)(X_{t+h} - X_t)^2.$

Step 2: For the Itô process $dX_t = \mu_t dt + \sigma_t dB_t$, compute $(dX_t)^2 = \sigma_t^2 dt + o(dt)$ using $dB_t^2 = dt$.

Step 3: Substitute into the Taylor expansion and take limits to obtain Itô's formula. The second-order term survives because of the non-zero quadratic variation.

Rigorously, this requires partitioning $[0,t]$ and showing convergence in probability.