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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

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Darboux sums are not mentioned because they are not useful in the definition of the stochastic integral. To see why, notice the difference between the Itō integral and the Stratonovich integral, in particular noting that the two give different answers.

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.

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My sincere thanks for this appreciable answer. By the way, I would like to know why we can't say:" The stochastic integral in itô sense is define like the Riemann integral's by using the Darboux sums" which pathological case we want to avoid. – Zbigniew Dec 30 '12 at 5:13
I added quite a bit of detail, which I hope answers your question. (I didn't have the time when I first answered the question.) – Harald Hanche-Olsen Dec 30 '12 at 14:13
After all, I'm satisfied for your reply. Respectfully. – Zbigniew Dec 30 '12 at 17:15

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