On Parisian ruin over a finite-time horizon

TL;DR

This paper derives asymptotic formulas for the probability of Parisian ruin over a finite horizon in a risk process with Gaussian claims, including tail asymptotics for Brownian motion with drift.

Contribution

It provides the first asymptotic expansion for Parisian ruin probability in Gaussian risk models over finite time horizons.

Findings

Asymptotic expansion of Parisian ruin probability as initial capital grows large.

Exact tail asymptotics for the infimum of Brownian motion with drift.

Extension of classical ruin results to finite-time Parisian setting.

Abstract

For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\}, \] with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.