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On first and last ruin times of Gaussian processes

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  • Hüsler, Jürg
  • Zhang, Yueming

Abstract

Considering centered Gaussian processes X(t) with a trend -ct[beta] and variance V2(t), we are interested in the asymptotic distributions of the first ruin time and the last ruin time as well as their joint asymptotic distribution as the initial capital u-->[infinity]. Our results show that the conditional distribution of the last ruin time, conditioned on ruin occurring, is a normal distribution and the conditional joint limit distribution is a difference of two standard normal distributions.

Suggested Citation

  • Hüsler, Jürg & Zhang, Yueming, 2008. "On first and last ruin times of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1230-1235, August.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:10:p:1230-1235
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    References listed on IDEAS

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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.
    2. 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.
    3. Zhang, Chunsheng & Wang, Guojing, 2003. "The joint density function of three characteristics on jump-diffusion risk process," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 445-455, July.
    4. Delbaen, Freddy, 1990. "A remark on the moments of ruin time in classical risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 121-126, September.
    5. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    6. Hüsler, Jürg & Piterbarg, Vladimir, 2004. "Limit theorem for maximum of the storage process with fractional Brownian motion as input," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 231-250, December.
    7. Debicki, Krzysztof, 2002. "Ruin probability for Gaussian integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 151-174, March.
    8. Egidio dos Reis, Alfredo D., 2000. "On the moments of ruin and recovery times," Insurance: Mathematics and Economics, Elsevier, vol. 27(3), pages 331-343, December.
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