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Topic Title: Variance of integrated Wiener process
Created On Tue Jun 20, 06 09:14 AM

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PiotrW
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Tue Jun 20, 06 09:14 AM

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Hello,

Mr Zhang in his paper �Theory of Continuously Sampled Asian Option Pricing� wrote during derivation of a formula for backward-starting geometric Asian option (page 6):

Perhaps someone knows why it is so.

Regards,
PiotrW

http://www.eaber.org/intranet/documents/23/228/CUHK_Zhang_00.pdf

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DavidF
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Tue Jun 20, 06 09:32 AM

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I'm sorry to reply you in words but I don't know anything about LaTex...

You have two ways to see that:
first you can just compute the expectation of the square of this integral, making it a double integral with Fubini's theorem (and use the property of the brownian motion that
E[WsWt]=min(s,t)
second you can apply Ito's lemma to the fonction (T-t-tau)*Wtau and it will tranform your integral to a stochastic integral

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PiotrW
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Fri Jun 23, 06 06:27 PM

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Thanks DavidF for an answer.

But do you know how to calculate it (there are two tau's as integration vatiables, and once you integrated internal integration there is no tau inside \int symbol for the second integration.

(basing on Fubini's theorem)

Regards,
PiotrW

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GoGoFa
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Mon Jun 26, 06 10:58 AM

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Of course you have two different variables of integration:

E[(\int W_t d t)^2] = E[\int W_s ds \int W_t dt] = E[\int\int W_s W_t dt ds] = \int\int E[W_sW_t] dt ds by Fubini
Now use E[W_sW_t]=min(s,t) and you're through.

PiotrW
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Thu Jun 29, 06 01:35 PM

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GoGoFa,

all right but what are the integration boundaries? Are they changed?

Regards,
PiotrW

PS. DavidF - oh, you have changed your 'face'

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ppauper
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Fri Jun 30, 06 03:27 PM

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hehehehe.... he said wiener

horacioaliaga
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Mon Jul 03, 06 05:57 PM

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This is what GoGoFa meant:

Edited: Mon Jul 03, 06 at 06:27 PM by horacioaliaga

PiotrW
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Tue Jul 04, 06 08:02 AM

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Thank you horacioaliaga.

Have a good day everybody.

PiotrW

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