First passage in an interval for fractional Brownian motion

Abstract

Be $X_t$ a random process starting at $x\in [0,1]$ with absorbing boundary conditions at both ends of the interval. Denote $P_1(x)$ the probability to first exit at the upper boundary. For Brownian motion, $P_1(x)=x$, equivalent to $P_1^{\prime }(x)=1$. For fractional Brownian motion with Hurst exponent $H$, we establish that $P_1^{\prime }(x)=\mathcal{N}[x(1-x){]}^{\frac{1}{H}-2}{e}^{\epsilon \mathcal{F}(x)+\mathcal{O}({\epsilon }^2)}$, where $\epsilon =H-\frac{1}{2}$. The function $\mathcal{F}(x)$ is analytic, and well approximated by its Taylor expansion, $\mathcal{F}(x)\simeq 16(C-1)(x-1/2{)}^2+\mathcal{O}(x-1/2{)}^4$, where $C=\mathrm{0.915...}$ is the Catalan-constant. A similar result holds for moments of the exit time starting at $x$. We then consider the span of $X_t$, i.e. the size of the (compact) domain visited up to time $t$. For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time, then generalized to fBm. Using large-scale numerical simulations with system sizes up to $N=2^{24}$ and a broad range of $H$, we confirm our analytic results. There are important finite-discretization corrections which we quantify. They are most severe for small $H$, necessitating to go to the large systems mentioned above.

21 pages, 54 figures. v2: additional material and clarifications added

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