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Given my earlier comment, the only open question is how $\frac{1}{2}\frac{d^2F(X(t))}{dX^2}\delta t$ becomes $\frac{1}{2}\int^{t+\delta t}_{t}\frac{d^2F(X(\tau))}{dX^2}d\tau\,.$ A more standard proof is this: Writing $$t_j=t+jh$$ we have \begin{align} &\sum_{j=1}^n\frac{d^2F}{dX^2}\Big(X(t_{j-1})\Big)\Big(X(t_j)-X(t_{j-1})\Big)^2\\ &=\sum_{j=1}^n\...