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

## TL;DR

This paper derives the exact asymptotics for the probability and timing of Parisian ruin over an infinite horizon in a Brownian motion risk model with exponential discounting and interest, providing new insights into risk process behavior.

## Contribution

It introduces the first precise asymptotic analysis of Parisian ruin probabilities and times for a Brownian motion-based risk model with interest and discounting.

## Key findings

- Exact asymptotics of Parisian ruin probability derived
- Asymptotic behavior of Parisian ruin time characterized
- Results applicable to risk models with interest and volatility

## 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^{\delta}(t)=e^{\delta t}\left(u+c\int^{t}_{0}e^{-\delta v}d v-\sigma\int_{0}^{t}e^{-\delta v}d B(v)\right),\quad t\geq0, \end{align} where $u\geq 0$ is the initial reserve, $\delta\geq0$ is the force of interest, $c>0$ is the rate of premium and $\sigma>0$ is a volatility factor. Further, we show the asymptotics of the Parisian ruin time of this risk process.