Persistence of fractional Brownian motion with moving boundaries and applications

TL;DR

This paper investigates the persistence probabilities of fractional Brownian motion with moving boundaries, establishing that logarithmic boundaries do not alter decay rates and applying these results to related stochastic properties.

Contribution

It demonstrates that adding a logarithmic boundary to FBM does not change the polynomial decay rate of persistence probability and extends these findings to continuous-time models and related quantities.

Findings

Persistence probability decay rate remains unchanged with logarithmic boundaries.

Results apply to continuous-time versions of related stochastic quantities.

Improves estimates for maximum and last zero times of FBM.

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. study a physical model where persistence properties of FBM are shown to be related to scaling properties of a quantity $J_N$, called 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 a FBM without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in Oshanin et al. Moreover, we complement their findings by considering the continuous-time version of $J_N$. Finally, we use the results for moving boundaries in order to improve estimates by Molchan concerning the persistence properties of other quantities of interest, such as the time when a FBM reaches its maximum on the time interval $(0,1)$ or the last zero in the interval $(0,1)$.