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

Long Bai

Scandinavian Actuarial Journal, 2018, vol. 2018, issue 6, 514-528

Abstract: Let B(t),t≥0$ B(t), t \ge 0 $ be a standard Brownian motion. In this paper, we derive the asymptotics of the probability of Parisian ruin over an infinite time horizon for the following risk process (0.1)Ruδ(t)=eδtu+c∫0te-δvdv-σ∫0te-δvdB(v),t≥0,$$ \begin{aligned} R_u^{\delta }(t)=e^{\delta t}\left(u+c\int ^{t}_{0}e^{-\delta v}\mathrm{d} v-\sigma \int _{0}^{t}e^{-\delta v}\mathrm{d} B(v)\right),\quad t \ge 0, \end{aligned} $$where u≥0$ u \ge 0 $ is the initial reserve, δ≥0$ \delta \ge 0 $ is the force of interest, c>0$ c>0 $ is the rate of premium and σ>0$ \sigma >0 $ is a volatility factor. It turns out that the Parisian ruin probability decays exponentially as u tends to infinity and is a decreasing function of the force of interest for u large. Moreover, we obtain the approximations of Parisian ruin time.

Date: 2018
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DOI: 10.1080/03461238.2017.1391872

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