An application that springs to mind immediately is *option evaluation*.
Suppose I offer you to buy a contract to me: after three months I pay you the maximum value of the price of an asset.
How much are you willing to pay this contract ?
If you model the price of the assert by a drifted brownian motion, then you'll probably want to estimate the distribution of this maximum and take the expected value as a first guess of this maximum price. This will also be my first guess at the minimum price at which I will be willing to sell the contract. This is a non arbitrage price.

Note to purists : of course one will object that each of the parties can hedge himself, and that the distribution has to be corrected by a risk neutral argument (which will probably discard the initial drift and replace it by a zero risk rate drift instead).

N.b. : this is a real world application, exotic option traders buy and sell like contracts everyday.

nottry to study the distribution of its supremum? (I assume you mean the process $Y_t = \sup_{o\leq s\leq t} X_s$.) I really don't understand what you mean by "application" and "project" $\endgroup$ – Yemon Choi Feb 8 '12 at 0:39