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kiann
Topic Author
Posts: 18
Joined: April 16th, 2008, 6:39 pm

### Derivation for two independent brownian motions

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

bearish
Posts: 6448
Joined: February 3rd, 2011, 2:19 pm

### Re: Derivation for two independent brownian motions

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.

Alan
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Joined: December 19th, 2001, 4:01 am
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### Re: Derivation for two independent brownian motions

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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Joined: August 8th, 2018, 9:10 am
Location: Singapore

### Re: Derivation for two independent brownian motions

@Alan,
The bonus question you provided is interesting!

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: Derivation for two independent brownian motions

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.

krs
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Re: Derivation for two independent brownian motions

is it not going to be max( F(x1), F(x2)....F(xn) )? when n is infinite, the value of the function will be equal to 1.

katastrofa
Posts: 10082
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Derivation for two independent brownian motions

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.
Curious animal. $\sqrt{-2\ln(1-\frac{1}{\sqrt[n]{2}})}$ would be the median?