First passage in an interval for fractional Brownian motion

Let Xₜ be a random process starting at x∈[0,1] with absorbing boundary conditions at both ends of the interval. Denote by P₁(x) the probability to first exit at the upper boundary. For Brownian motion, P₁(x)=x, which is equivalent to P₁^'(x)=1. For fractional Brownian motion with Hurst exponent H, we establish that P₁^'(x)=N[x(1-x)]^1/H-2e^εF(x)+O(ε²), where ε=H-1/2. The function F(x) is analytic and well approximated by its Taylor expansion F(x)16(C-1)(x-1/2)²+O(x-1/2)⁴, where $C=0.915...$ is the Catalan constant. A similar result holds for moments of the exit time starting at x. We then consider the span of Xₜ, 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 and then generalize it to fractional Brownian motion. Using large-scale numerical simulations with system sizes up to N=2²⁴ 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 going to the large systems mentioned above.