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jmmm465
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Joined: May 7th, 2009, 7:35 pm

Having trouble with part of a Brownian Motion / Martingale exercise

May 7th, 2009, 9:33 pm

Hello! I just found these wonderful forums. I'm self studying with Volume I and II of Shreve.I'm currently studying the material in Chapter 3 of Volume 2. Here is the problem I am working on:Let W(t) be a brownian motion, let F(t) be a filtration for this brownian motion, and 0 <= t. Show that W^3(t) - 3tW(t) is a martingale.I found that E[W^3(t) | F(t)] = W^3(s).I need to show that E[-3tW(t) | F(t)] = -3sW(s), for 0 <= s <= t and i'll be done. The 't' is bothering me... I know I can pull the -3 out of the Expectation, but I'm having trouble showing E[tW(t)|F(t)] = sW(s). It's clear to me that E[W(t) | F(t)] = W(s)... I'm just wondering what property I'm missing in regards to the t inside the expectation with the given filtration. Thanks for your help, this is probably a pretty trivial problem.
 
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QuantOption
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Having trouble with part of a Brownian Motion / Martingale exercise

May 8th, 2009, 12:16 pm

QuoteI found that E[W^3(t) | F(t)] = W^3(s).wrongQuoteI need to show that E[-3tW(t) | F(t)] = -3sW(s)wrong
 
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jmmm465
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Joined: May 7th, 2009, 7:35 pm

Having trouble with part of a Brownian Motion / Martingale exercise

May 8th, 2009, 12:24 pm

Thanks for the response.I really did have a feeling the second expectation was wrong... which leads to the first being wrong. I'll revisit my notes on the exercise when I get home from work today and try again. My approach was to write W(t) as (W(t) - W(s)) + W(s)... After a significant amount of algebraic manipulation and some properties about brownian motions from the text, I ended up with W^3(s) for the expectation of W^3(t)... written as E[(W(t) - W(s))^3 + W(s)] EDIT: I'm still going to have trouble based on my original question (even if some of my other work is wrong too, hah). I'm not sure how to handle the 't' inside the expectation... I don't think I can just pull it out like a constant (i.e. E[-3tW(t)] = -3tE[W(t)]). Any tips?
Last edited by jmmm465 on May 7th, 2009, 10:00 pm, edited 1 time in total.
 
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bilbo1408
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Having trouble with part of a Brownian Motion / Martingale exercise

May 8th, 2009, 4:29 pm

First....
Last edited by bilbo1408 on May 8th, 2009, 10:00 pm, edited 1 time in total.