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Ruin probability in the presence of risky investments

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  • Pergamenshchikov, Serguei
  • Zeitouny, Omar

Abstract

We consider an insurance company in the case when the premium rate is a bounded non-negative random function ct and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a and volatility [sigma]>0. If [beta]:=2a/[sigma]2-1>0 we find exact the asymptotic upper and lower bounds for the ruin probability [Psi](u) as the initial endowment u tends to infinity, i.e. we show that C*u-[beta][less-than-or-equals, slant][Psi](u)[less-than-or-equals, slant]C*u-[beta] for sufficiently large u. Moreover if ct=c*e[gamma]t with [gamma][less-than-or-equals, slant]0 we find the exact asymptotics of the ruin probability, namely [Psi](u)~u-[beta]. If [beta][less-than-or-equals, slant]0, we show that [Psi](u)=1 for any u[greater-or-equal, slanted]0.

Suggested Citation

  • Pergamenshchikov, Serguei & Zeitouny, Omar, 2006. "Ruin probability in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 267-278, February.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:2:p:267-278
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    References listed on IDEAS

    as
    1. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
    2. Paulsen, Jostein, 1998. "Sharp conditions for certain ruin in a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 135-148, June.
    3. Kalashnikov, Vladimir & Norberg, Ragnar, 2002. "Power tailed ruin probabilities in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 211-228, April.
    4. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
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    Citations

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    Cited by:

    1. Yuri Kabanov & Serguei Pergamenshchikov, 2020. "Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process," Finance and Stochastics, Springer, vol. 24(1), pages 39-69, January.
    2. Xiang Lin, 2009. "Ruin theory for classical risk process that is perturbed by diffusion with risky investments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 33-44, January.
    3. Henrik Hult & Filip Lindskog, 2011. "Ruin probabilities under general investments and heavy-tailed claims," Finance and Stochastics, Springer, vol. 15(2), pages 243-265, June.
    4. Xiong, Sheng & Yang, Wei-Shih, 2011. "Ruin probability in the Cramér-Lundberg model with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1125-1137, May.
    5. Yuri Kabanov & Sergey Pergamenshchikov, 2022. "On ruin probabilities with investments in a risky asset with a regime-switching price," Finance and Stochastics, Springer, vol. 26(4), pages 877-897, October.
    6. Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
    7. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.

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