# Distribution of running maximums of a log normal process

I've been searching for quite some time and would appreciate any guidance! What I'm looking for is the distribution of running maximums for a log-normal process. If anyone is familiar with any relevant material, please point me in the right direction. Thanks.

• A very warm welcome to Quant.SE! What do you mean by "running maxima"? (maxima is the plural of maximum.) – vonjd Apr 21 '15 at 18:20
• Thanks @vonjd! I may have the terminology wrong, but when I say "running maximum," I mean something along the lines of: mathworld.wolfram.com/High-WaterMark.html. For a log normal process of some length, I'm wondering if there is an analytical solution describing the distribution of the largest value in the process. Does this make sense? – cjken Apr 21 '15 at 18:37
• Here is a sketch. 1) For a brownian motion the law of the running max is determined by the law of the hitting time which is computed using the reflexion principle. 2) To include a drift; use Girsanov to reduce to 1 and then Girsanov again to get the distribution under the original measure. 3) exp(X_t) reaches $H$ iff $X_t$ reaches $\log H$ so you just need to make a change of variable in the previous result to get the result for GBM. A general lognormal process could be more complicated. – AFK Apr 22 '15 at 5:14