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Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

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  • Yuri Kabanov

    (Université de Franche-Comté
    Lomonosov Moscow State University
    Federal Research Center “Computer Science and Control” of Russian Academy of Sciences)

  • Serguei Pergamenshchikov

    (Université de Rouen
    National Research Tomsk State University)

Abstract

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Lévy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let β>0$\beta >0$ be the root of the cumulant-generating function H$H$ of the increment V1$V_{1}$ of the log-price process. We show that the ruin probability admits the exact asymptotic Cu−β$Cu^{-\beta }$ as the initial capital u→∞$u\to \infty $, assuming only that the law of VT$V_{T}$ is non-arithmetic without any further assumptions on the price process.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:1:d:10.1007_s00780-019-00413-3
    DOI: 10.1007/s00780-019-00413-3
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    References listed on IDEAS

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    2. Erhan Bayraktar & Masahiko Egami, 2008. "Optimizing venture capital investments in a jump diffusion model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 21-42, February.
    3. 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.
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    9. 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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    Cited by:

    1. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Leshchinskaya, 2022. "Improved estimation method for high dimension semimartingale regression models based on discrete data," Statistical Inference for Stochastic Processes, Springer, vol. 25(3), pages 537-576, October.
    2. Yuri Kabanov & Platon Promyslov, 2023. "Ruin probabilities for a Sparre Andersen model with investments: the case of annuity payments," Finance and Stochastics, Springer, vol. 27(4), pages 887-902, October.
    3. Eberlein, Ernst & Kabanov, Yuri & Schmidt, Thorsten, 2022. "Ruin probabilities for a Sparre Andersen model with investments," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 72-84.
    4. Andreas Karathanasopoulos & Chia Chun Lo & Xiaorong Ma & Zhenjiang Qin, 2021. "Maintaining cost and ruin probability," Review of Quantitative Finance and Accounting, Springer, vol. 57(2), pages 759-793, August.

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    More about this item

    Keywords

    Ruin probabilities; Dual models; Price process; Renewal theory; Distributional equation; Autoregression with random coefficients; Lévy process;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors

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