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Parisian ruin of the Brownian motion risk model with constant force of interest

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  • Bai, Long
  • Luo, Li

Abstract

Let B(t),t∈R be a standard Brownian motion. Define a risk process (0.1)Ruδ(t)=eδt(u+c∫0te−δsds−σ∫0te−δsdB(s)),t≥0, where u≥0 is the initial reserve, δ≥0 is the force of interest, c>0 is the rate of premium and σ>0 is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability KSδ(u,Tu):=P{inft∈[0,S]sups∈[t,t+Tu]Ruδ(s)<0},S≥0, as u→∞ where Tu 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 Tu≡0 in the Parisian setting.

Suggested Citation

  • Bai, Long & Luo, Li, 2017. "Parisian ruin of the Brownian motion risk model with constant force of interest," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 34-44.
    • Handle: RePEc:eee:stapro:v:120:y:2017:i:c:p:34-44
    DOI: 10.1016/j.spl.2016.09.011
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    1. Hüsler, J. & Piterbarg, V., 1999. "Extremes of a certain class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 257-271, October.
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    3. Dieker, A.B., 2005. "Extremes of Gaussian processes over an infinite horizon," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 207-248, February.
    4. Griselda Deelstra, 1994. "Remarks on Boundary crossing..," ULB Institutional Repository 2013/7576, ULB -- Universite Libre de Bruxelles.
    5. Harrison, J. Michael, 1977. "Ruin problems with compounding assets," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 67-79, February.
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    Cited by:

    1. Peng, Xiaofan & Luo, Li, 2017. "Finite time Parisian ruin of an integrated Gaussian risk model," Statistics & Probability Letters, Elsevier, vol. 124(C), pages 22-29.
    2. Krystecki, Konrad, 2022. "Parisian ruin probability for two-dimensional Brownian risk model," Statistics & Probability Letters, Elsevier, vol. 182(C).

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