The purpose of this note is to obtain a representation of the distribution of the $\alpha$-quantile of a process with stationary and independent increments as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for a Brownian motion. The proof is based on already known discrete time results.
"Sample quantiles of stochastic processes with stationary and independent increments." Ann. Appl. Probab. 6 (3) 1041 - 1043, August 1996. https://doi.org/10.1214/aoap/1034968241