First passage in an interval for fractional Brownian motion

Let X_{t} be a random process starting at x∈[0,1] with absorbing boundary conditions at both ends of the interval. Denote by P_{1}(x) the probability to first exit at the upper boundary. For Brownian motion, P_{1}(x)=x, which is equivalent to P_{1}^{'}(x)=1. For fractional Brownian motion with Hurst exponent H, we establish that P_{1}^{'}(x)=N[x(1-x)]^{1/H-2}e^{εF(x)+O(ε^{2})}, 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 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 and then generalize it to fractional Brownian motion. 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 going to the large systems mentioned above.