Where the integrand is read inside each step changes the answer. Left-point sums give the Itô integral (W_T² − T)/2; midpoint sums give the Stratonovich value W_T²/2.
Path
1.00
0.255
Random seed: 7
Itô: (W_T² − T)/2
0.998
Stratonovich: W_T²/2
1.498
Left point (Itô sum), n = 2048
0.981
Midpoint − left, mean of 100 paths
0.503 (T/2 = 0.500)
Right − left, mean of 100 paths
1.004 (T = 1.000)
Right − left is the sum of squared increments, which tends to T; the midpoint sits half way, so its gap is T/2.
Riemann sums of ∫W dW against the number of steps
Left point (Itô sum)MidpointRight pointItô: (W_T² − T)/2Stratonovich: W_T²/2