Persistence of fractional Brownian motion with moving boundaries and applications

Abstract

We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM X stays below a certain level until time T. Recently, Oshanin et al (2012, arXiv:1209.3313v2) have studied a physical model, where persistence properties of FBM are shown to be related to scaling properties of a quantity JN, called the steady-state current. It turns out that for this analysis, it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to an FBM without changing the polynomial rate of decay of the corresponding persistence probability, which proves a result needed in Oshanin et al (2013 Phys. Rev. Lett. at press (arXiv:1209.3313v2)). Moreover, we complement their findings by considering the continuous-time version of JT. Finally, we use the results for moving boundaries in order to improve estimates by Molchan (1999 Commun. Math. Phys. 205 97–111) concerning the persistence properties of other quantities of interest, such as the time when an FBM reaches its maximum on the time interval (0, 1) or the last zero in the interval (0, 1).