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Postby Lauri » Thu Sep 04, 2008 8:12 pm

The answer for why Brownian motion is nowhere differentiable is easy. It is because of its nonvanishing quadratic variation. Note that the nonvanishing quadratic variation is necessary requirement for the function to be differentiable.

It is quite easy to show the quadratic variation of the Brownian motion to be b-a in any finite interval [a,b]
Lauri
 
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Postby mj » Fri Sep 05, 2008 12:59 am

that's the idea but to cross all the ts and dot all the i's is tricky.

eg with the standard definition of quadratic variation, (qv), it is actually infinite over any interval [a,b] with probability 1. So you have to use a different definition.

The statement that the qv on a given interval [a,b] is b-a with probability 1, is not the same as the statement the qv on every interval [a,b] is b-a with probability 1.
mj
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Postby Muble » Sun Sep 07, 2008 3:56 pm

As Mark says, it is indeed very difficult to cross the t's and the i's and the bare hands proof of non differentiabilty is very tough.

However I found in Karatzas & Schreve a very neat somewhat advanced shortcut that uses local time. Anyone interested, check Exercise 6.6 pg 203
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