The Itô Integral for Dummies

What to remember

• Left sum $\sum W(t_i)\,\Delta W_i \to \tfrac12 W_T^2 - \tfrac12 T$; right sum $\sum W(t_{i+1})\,\Delta W_i \to \tfrac12 W_T^2 + \tfrac12 T$.
• The gap between them is $\sum (\Delta W_i)^2$, the quadratic variation, which tends to $T$ instead of $0$.
• The Itô integral uses the left endpoint: the integrand is fixed before the increment, like a position set before the price moves.
• $E[\int W\,dW] = 0$ and $\mathrm{Var}(\int_0^T W\,dW) = T^2/2$.
• Averaging the two endpoints gives the Stratonovich integral, $\tfrac12 W_T^2$, which keeps the ordinary chain rule and gives up the martingale property.
• Brownian motion has no slope at any scale, so $\int W\,dW$ cannot become $\int W\,W'\,dt$; it is defined directly as a limit of sums, and $W_T^2 = 2\sum W\,\Delta W + \sum (\Delta W)^2$ holds exactly on every partition.