Consider a stochastic differential equation such as $dX = C\, dt + \sigma\, dB$ With a condition such as $X(0)=0$. I consider this variable across the time interval $[0,1]$. Calculating the distribution of X(1) is trivial, but I'm trying to understand the behavior of min(X) and max(X). (I believe this is from what is known as extremal calculus, but I don't know any and couldn't find a simple reference.) I do know of course that from the odd time symmetry, the distributions follow
$\min(X) \sim \mu - \max(X)$
Now while I'm sure that this is a solved problem, I'm curious how the correlations between the minimum and maximum behave. I expect that for large mu (equiv: large time) they become less and less correlated.