Stochastic ordering for hitting times of fractional Brownian motions

Abstract

In this note we prove that, under a suitable transformation, the hitting times of fractional Brownian motions (fBMs) can be ordered stochastically. As a consequence, we derive upper and lower bounds for their cumulative distribution function. We also study some properties, such as continuity and monotonicity, of the probability tail and of the expectation of the supremum of fBMs seen as functions of the Hurst index H. We apply these properties to provide alternative proofs of some results known for fBM.