Sojourn Ruin of a Two-Dimensional Fractional Brownian Motion Risk Process

Abstract

This paper derives the asymptotic behavior of $$\mathbb{P}\{\underset{0}{\overset{\infty }{\int }}\mathbb{I}(B_H(s)-c_{1}s>q_{1}u,B_H(s)-c_{2}s>q_{2}u)ds>T_u\},u\to \infty ,$$ where $B_H$ is a fractional Brownian motion, $c_1,c_2,q_1,q_2>0,H\in (0,1),T_u\ge 0$ is a measurable function and $\mathbb{I}(\cdot )$ is the indicator function.