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#1
04-08-2007, 05:05 PM
 KingWithoutACrown Member Join Date: Apr 2006 Posts: 1,510

What are the significance of these concepts?
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#2
04-09-2007, 08:29 PM
 Captain Nemo Bill Cross Join Date: Aug 2004 Posts: 1,129

Quote:
 Originally Posted by KingWithoutACrown What are the significance of these concepts?

There are some interesting theorems in functional analysis regarding the variation of functions (for example, if a function has finite total variation over any finite interval, then it can be expressed as the difference of two monotone nondecreasing functions over that interval) but other than referencing "total variation", "quadratic variation", and "infinite crossing property", I don't think it's all that testable.

"Which of the following statements is true:

I. If Z(t) is Brownian motion, then Z(t) has finite expected quadratic variation.

II. If Z(t) is Brownian motion, then Z(t) has finite expected total variation.

III. If Z(t) is Brownian motion, and Z(3) = 2, then the expected number of values of t between 3 and 5 such that Z(t) = 2 is equal to 2^(0.5).

(A) I
(B) II
(C) I and III
(D) II and III
(E) None of the above.
"

(A)
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#3
08-31-2012, 01:11 PM
 Londondrug Member CAS SOA Join Date: Apr 2012 Posts: 728

#4
08-31-2012, 01:23 PM
 AAABBBCCC Member Join Date: Feb 2012 Posts: 1,329

Hes talking about the expected measure of the zero set of brownian motion over a compact interval.

LOL
#5
08-31-2012, 01:27 PM
 Londondrug Member CAS SOA Join Date: Apr 2012 Posts: 728

Quote:
 Originally Posted by AAABBBCCC Hes talking about the expected measure of the zero set of brownian motion over a compact interval. LOL
I know. But I have no clue how to judge II and III, especially III.
#6
08-31-2012, 01:29 PM
 AAABBBCCC Member Join Date: Feb 2012 Posts: 1,329

which is zero, but I dont know what the expected Hausdorff dimension is.
#7
08-31-2012, 01:29 PM
 Londondrug Member CAS SOA Join Date: Apr 2012 Posts: 728

For II, I guess is wrong, because the first order of variation of Z(t) is infinite.
#8
08-31-2012, 01:32 PM
 AAABBBCCC Member Join Date: Feb 2012 Posts: 1,329

Quote:
 Originally Posted by Londondrug I know. But I have no clue how to judge II and III, especially III.
statement III makes no sense (as it stands). it says how many values hit 2 and claims that it's the square root of 2. That doesnt make any sense.

Hes talking about the expected measure of the set of points {t: z(t)=2, 3<=t<=5}, which I believe is zero. It is just fubini's theorem.

Last edited by AAABBBCCC; 08-31-2012 at 01:39 PM..
#9
08-31-2012, 01:33 PM
 AAABBBCCC Member Join Date: Feb 2012 Posts: 1,329

Quote:
 Originally Posted by Londondrug For II, I guess is wrong, because the first order of variation of Z(t) is infinite.
Yes, it is wrong. Brownian paths are differentiable nowhere with probability one while functions of bounded variation are differentiable almost everywhere.
#10
08-31-2012, 01:53 PM
 Londondrug Member CAS SOA Join Date: Apr 2012 Posts: 728

Many thanks, AAABBBCCC.
probably III is trying to say the variance of Z(5)-Z(3) is 2, then the sigma is 2^0.5, however, it has nothing to do with Quadratic Variation.

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