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arXiv · 1301.0424

Persistence of fractional Brownian motion with moving boundaries and applications

Frank Aurzada·Christoph Baumgarten
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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 $(0,1)$.

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Frank Aurzada, Christoph Baumgarten. 2013-01-03. Persistence of fractional Brownian motion with moving boundaries and applications. https://doi.org/10.1088/1751-8113%2F46%2F12%2F125007

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