Why can the stochastic integral $\int_{t_i}^{t_{i+1}} \sigma(t,X(t))dW(t) $ be approximated by $\sigma(t_i,X(t_i))(W(t_{i+1})-W(t_i))$ in the Euler scheme? Here $\sigma$ can be thought of as a lipschitz continuous bounded function and $W$ as the standard one-dimensional Brownian motion
If this were a path by path integral($\omega$-wise in the Lebesgue Stieltges sense) , I would have no apprehensions what so ever. But the stochastic integral isnt defined path-wise. Does is have to do with the way we define stochastic integrals of simple functions with respect to the Brownian motion?