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### Maximum of Brownian Motion

Posted: **August 2nd, 2011, 6:33 pm**

by **DoubleTrouble**

Hi!I would like to present a solution for how to calculate where and I'd really appreciate if you guys would be so kind to pointed out where my reasoning has flaws (if it has any).First of all we need to find the probability density function of . To do this we use the Reflection Principle which tells us that and in other words . Which tells us that the density function is And now the expectation can be calculated:Is this reasoning valid?Thank you in advance!

### Maximum of Brownian Motion

Posted: **August 2nd, 2011, 6:37 pm**

by **DoubleTrouble**

Perhaps one can find the distribution of easier by reasoning like this:where

### Maximum of Brownian Motion

Posted: **August 2nd, 2011, 7:29 pm**

by **ThinkDifferent**

QuoteOriginally posted by: DoubleTroublePerhaps one can find the distribution of easier by reasoning like this:where the approach in your first post is a standard one ( i didn't check for exact calculations). what you are doing in the second post doesn't seem to lead anywhere. in fact, starting from the second equality sign it becomes nonsense.... The confusion comes from the fact that W_t is not equal to sqrt(t) N. Both have the same distribution though.

### Maximum of Brownian Motion

Posted: **August 2nd, 2011, 7:49 pm**

by **DoubleTrouble**

QuoteOriginally posted by: ThinkDifferentwhat you are doing in the second post doesn't seem to lead anywhere. in fact, starting from the second equality sign it becomes nonsense.... Yeah, I see why now! Thanks for your input!

### Maximum of Brownian Motion

Posted: **August 3rd, 2011, 5:06 am**

by **mirthgrief**

Hey I wonder if you could name 5 to 6 physics-related models other than the brownian motion models that is used in quantitative finance?Also, do these models work well in extreme, non-random situations such as the 2008 subsidary mortgage crisis or in other financial turmoils?

### Maximum of Brownian Motion

Posted: **August 21st, 2011, 7:44 pm**

by **DoubleTrouble**

I realized today that I made a slight error in my first post. That density formula holds only for so the integral should range from 0 to infinity which gives the previous result divided by 2.Right?