The first exit time of fractional Brownian motion from an unbounded domain

Abstract

Consider a fractional Brownian motions starting at the interior point $\left(x_0,h\left(\Vert x_0\Vert \right)+2K\right)\in R^{d+1}$ with the constant $K>1$, for some fixed $x_0\in R^d$, of an unbounded domain $D=\left(\left(x,y\right)\in R^{d+1}:y>h\left(\Vert x\Vert \right)\right)$, The function $h$ is a nondecreasing, lower semicontinuous, and convex function on $[0,\infty )$ with $h\left(0\right)$ being finite. Here we take $h^{-1}\left(x\right)=A{x}^{\alpha }{\left(logx\right)}^{\beta }$with a positive constant $A$ for $x>K$. It is evident that $h^{-1}\left(x\right)$ exhibits monotonic behavior for sufficiently large values of $x$. Let ${\tau }_D$ denote the first time that the fractional Brownian motion exits from $D$. In most cases, we give the asymptotically equivalent estimate of $logP\left({\tau }_D>t\right)$. The proof methods are based on the earlier works of Li, Shi, Lifshits, and Aurzada.