Parisian Ruin of the Brownian Motion Risk Model with Constant Force of Interest

TL;DR

This paper derives asymptotic approximations for the Parisian ruin probability and ruin time of a Brownian motion risk model with constant interest, extending classical ruin results to the Parisian setting.

Contribution

It provides the first asymptotic analysis of Parisian ruin probability and ruin time for a Brownian motion risk model with interest, including the case when the Parisian delay is zero.

Findings

Approximate Parisian ruin probability as initial reserve grows large.

Ruin time can be approximated by an exponential distribution.

Results extend classical ruin probability to the Parisian context.

Abstract

Let $B(t), t\in \mathbb{R}$ be a standard Brownian motion. Define a risk process \label{Rudef} R_u^{\delta}(t)=e^{\delta t}\left(u+c\int^{t}_{0}e^{-\delta s}d s-\sigma\int_{0}^{t}e^{-\delta s}d B(s)\right), t\geq0, 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. In this contribution we obtain an approximation of the Parisian ruin probability \mathcal{K}_S^{\delta}(u,T_u):=\mathbb{P}\left\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u^{\delta}(s)<0\right\}, S\ge 0, as $u\rightarrow\infty$ where $T_u$ is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to $T_u\equiv0$ in the Parisian setting.