The study of the quantile of a Brownian motion with a drift is undertaken. An explicit formula for its density, as well as a representation of its distribution as the sum of the maximum and the minimum of two rescaled independent Brownian motions with drift, is given. The result is used in the pricing of a financial path-dependent option due to Miura.
"The Distribution of the Quantile of a Brownian Motion with Drift and the Pricing of Related Path-Dependent Options." Ann. Appl. Probab. 5 (2) 389 - 398, May, 1995. https://doi.org/10.1214/aoap/1177004770