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

TL;DR

This paper analyzes the asymptotic probability of a two-dimensional fractional Brownian motion risk process exceeding a certain ruin threshold over time, providing insights into its long-term behavior.

Contribution

It derives the asymptotic behavior of the ruin probability for a two-dimensional fractional Brownian motion risk process, extending existing models to fractional Brownian motion.

Findings

Asymptotic formulas for ruin probabilities as u approaches infinity.

Characterization of the impact of Hurst parameter on ruin behavior.

Extension of classical ruin theory to fractional Brownian motion context.

Abstract

This paper derives the asymptotic behavior of $$\mathbb{P} \{ \int\limits_0^\infty \mathbb{I}\Big(B_H(s)-c_1s>q_1u, B_H(s)-c_2s>q_2u\Big)ds>T_u\},\quad u \to \infty,$$ 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.