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joint law of minimum and maximum
Would someone have a reference about the joint law of the maximum and the minimum of a drifted brownian motion in a finite time interval.I know there are formulae for the law of the maximum, of the minimum and for the joint law of the maximum (or minimum) and the brownian motion itself.But I don't find any reference for what I'm looking for. Any idea?Thanks
joint law of minimum and maximum
I haven't seen that one either -- if, by chance, all you need is reallythe "range of a brownian motion", google will show a lot of results on that one.regards,
joint law of minimum and maximum
Thank you for these answers.I'm definitely interested in the joint law of the min and the max of a drifted BM. I also thought to the book of borodin, but unfortunately I haven't access to it.Actually, my question was poorly formulated. More than a reference, I need the formula! If someone is courageous enough to write it, I will really appreciate. Otherwise a reference to an article accessible through the web would be perfect.Thanks for all!
joint law of minimum and maximum
Using EngMoi's excellent idea, here it is for no drift:(from B & S, 2nd ed. page 173)where the B.M. starts at x, has mininum m_t and maximum M_tregards, p.s. I noticed after typing, that you wanted a drift. With drift \mu, the only changes are (p.270): multiply the integrand bye^{-\mu^2 t/2 + \mu (z-x)}... eqn editor not working (hope you can read it)Now that I see it, I realize it's just the solution to the heat equationwith absorption at (a,b), which you get by the method of images. In finance, of course,this is just a double barrier ko option.
Last edited by Alan on March 5th, 2004, 11:00 pm, edited 1 time in total.