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kiann
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Joined: April 16th, 2008, 6:39 pm

Derivation for two independent brownian motions

March 30th, 2019, 10:19 pm

Suppose W1(t) and W2(t) are two independent standard Brownian motions. What is the probability that both processes are larger than 0.667 at t=1.0
 
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bearish
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Re: Derivation for two independent brownian motions

March 31st, 2019, 2:47 am

So you have two independent random variables, W1(1) and W2(1), each of which are distributed as N(0,1). The probability of both being larger than x is going to be the product of each being greater than x. 
 
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Alan
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Re: Derivation for two independent brownian motions

April 1st, 2019, 3:27 am

Here's a standard related problem that's a good follow-up for the OP to puzzle over. 

Consider a random sequence: the location of n independent BM's at time 1. 
What's the distribution function of the maximum of the sequence; 
i.e., what's [$]F(x)[$] where

[$]F(x) =  Prob \left( \max \left\{W_1(1), W_2(1), \cdots, W_n(1) \right\} < x \right)[$]  ?

Bonus: what's the expected value of that maximum as n becomes very large (but not infinite)?  

Hint: given bearish's comment, the first answer is easy and the second is hard (if you don't google).
 
galvinator
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Re: Derivation for two independent brownian motions

July 30th, 2019, 6:28 am

@Alan,
The bonus question you provided is interesting! 
I have not googled but would the answer be 0?
 
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Alan
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Re: Derivation for two independent brownian motions

July 30th, 2019, 11:31 pm

Thanks.

No, first, I said "n is large but not infinite", so the puzzle is to provide an asymptotic formula that depends upon n. Keep thinking!

Second, if n were infinite, the expected max is [$]+\infty[$].

Hint: start from bearish's comment, generalize it, and start calculating. 
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