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The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options. (English) Zbl 0837.60076
Let $$X$$ be Brownian motion with a constant drift and $$M(\alpha, t)$$ the $$\alpha$$-quantile of $$X$$ up to time $$t$$, i.e., the smallest value $$x$$ for which the time spent by $$X$$ below $$x$$ before $$t$$ exceeds $$\alpha t$$. The main result of the paper is that $$M(\alpha, t)$$ has the same distribution as the sum of the maximum of $$X$$ up to time $$\alpha t$$ and the minimum of an independent copy of $$X$$ up to time $$(1- \alpha) t$$. This is proved analytically by computing the Laplace transform of $$M(\alpha, t)$$ with the help of the Feynman-Kac formula. The same identity in law has recently been proved directly by P. Embrechts, L. C. G. Rogers and M. Yor [ibid. 5, No. 3, 757-767 (1995)], using more probabilistic arguments. As an application of the basic result, the author recovers a formula by Miura and Akahori for the price of a path- dependent financial option.

##### MSC:
 60J65 Brownian motion
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