# Min and Max of Geometric Brownian motion

I am trying to derive the distribution of $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$ and $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$, where $dX(t)=\mu X(t) dt+\sigma X(t)dB(t)$ and $B(t)$ is standard Brownian motion. I have seen the derivation of only Brownian motion i.e., $X(t) =\mu dt+\sigma dB(t)$ based on the reflection principle. I am not looking for something rigourous but just reflection principle and change of measure. A diagram will be of much help.

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$\exp$ is monotone and continuous. –  cardinal Jan 13 '12 at 14:03
@cardinal, yes how do I proceed –  Vaolter May 18 '12 at 10:12