Starting point

Write

$$ f(B_T)-f(B_0)

\sum_{i=0}^{N-1}\bigl(f(B_{t_{i+1}})-f(B_{t_i})\bigr). $$

Taylor expansion

For each increment,

$$ f(B_{t_{i+1}})-f(B_{t_i})

f’(B_{t_i})\Delta B_i + \frac12 f’’(B_{t_i})(\Delta B_i)^2 + O(|\Delta B_i|^3), $$

where

$$ \Delta B_i=B_{t_{i+1}}-B_{t_i}. $$

What survives in the limit

  • The first-order sum becomes the Ito integral
$$ \int_0^T f'(B_t)\,dB_t. $$
  • The second-order sum survives because
$$ (\Delta B_i)^2\approx \Delta t. $$
  • The third-order remainder disappears because it is of smaller order overall.

Core message

Ito formula is Taylor expansion plus the fact that Brownian quadratic variation does not vanish.