Parisian ruin of Brownian motion risk model over an infinite-time horizon

Abstract

Let $B(t),t\in \mathbb{R}$ be a standard Brownian motion. In this paper, we derive the exact asymptotics of the probability of Parisian ruin on infinite time horizon for the following risk process \begin{align}\label{Rudef} R_u^δ(t)=e^{δt}\left(u+c\int^{t}_{0}e^{-δv}d v-σ\int_{0}^{t}e^{-δv}d B(v)\right),\quad t\geq0, \end{align} where $u\ge 0$ is the initial reserve, $\hat{I}´\ge 0$ is the force of interest, $c>0$ is the rate of premium and $\ddot{I}ƒ>0$ is a volatility factor. Further, we show the asymptotics of the Parisian ruin time of this risk process.

10 pages