# Darboux versus stochastic integral

I don't know if my question is obscure. I'm astonished why there not mention the Darboux sums in the definition of stochastic integral

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The end of the world does'nt occurred. – Zbigniew Dec 30 '12 at 4:49

Edited to add more detail: You can think of the Itō integral as defined using Riemann–Stieltjes sums with the integrand always evaluated at the left end of each subinterval, whereas the Stratonivich integral uses instead the average of values at the left and right ends of each subinterval. Darboux sums, in contrast, use the minimum and maximum values of the integrand on each subinterval. Therein lies a problem of its own, for Brownian motion is not monotone, and so $B(t_{i+1})-B(t_i)$ can have either sign. You might remedy that by defining the upper Darboux sum to use the maximum in intervals where $B(t_{i+1})-B(t_i)$ is positive, and the minimum where it is negative – and the opposite for the lower Darboux sum. In which case the Itō sum and Stratonivich sums will both lie between the upper and lower Darboux sums. Since the former two will in general converge to different values, so must the latter two, and hence you cannot define the Darboux integral in any meaningful sense. The problem, it seems, is that the integrands which interest us have too much local variation for the Darboux approach to work.