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Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is said to be the chain rule for stochastic processes.

My questions: Some purists tell you that only the integral form is valid and the differential form is a shortcut to that at most. Why do they think so? Others tell you that both forms are ok and are just two sides of the same coin. What is true now and why?

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0 vote down The differential form is just a suggestive shorthand for the integral form. It's just as valid as the integral form, because it is the same thing. Do you have any reference which suggests otherwise? – George Lowther Oct 29 '09 at 20:31
Maybe we could rephrase the question a bit. In calculus, students learn about differentials as similar shorthand form. But in differential geometry, one learns that differentials have a life of their own as proper mathematical objects, given the right definition. So the rephrased question is if the same is true here? Is there a rigourous theory of stochastic differentials that includes dB, where B is Brownian motion? – Harald Hanche-Olsen Oct 29 '09 at 20:43
A reference would be Don M. Chance:… There he states (p.5): "Remember that either the differential or integral version of Itô’s Lemma automatically implies that the other exists, so either can be used, and in some cases, one is preferred over the other." – vonjd Oct 29 '09 at 21:02
...I mean a reference that both are the same thing - but what do you mean by "suggestive shorthand for the integral form"? Why isn't just a differential equation like other differential equations? Or are the non-stochastic ones also only "suggestive shorthands"? – vonjd Oct 29 '09 at 21:07
Things like Brownian motion aren't pointwise differentiable. Stochastic differential equations are normally (always?) defined via the integral form. – George Lowther Oct 29 '09 at 21:41
up vote 6 down vote accepted

I think there's an issue with definitions here. Ito's lemma in differential form only makes sense in the context of the integral form. It makes absolutely no sense to speak of dW(s)/ds, where W is Brownian motion since it's nowhere differentiable. We DEFINE dXt = sdB_t+mdt by saying that this is shorthand for X(t)-X(0)= int_0^t s dB_s + int_0^t m ds . To reiterate, the differential form is a shorthand way of writing the integral form. It's like the notion of defining weak derivatives. You can't speak with a straight face about the derivative (in the usual sense) of say, a dirac delta function, but you can define it through an integral.

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Maybe this book will help you. In some sense it gives a honest life to dW^2=dt. But "it is another story". Nonstandard methods in stochastic analysis and mathematical physics

See for example page 113.

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