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The Cramér-Lundberg model with a fluctuating number of clients

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  • Braunsteins, Peter
  • Mandjes, Michel

Abstract

This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.

Suggested Citation

  • Braunsteins, Peter & Mandjes, Michel, 2023. "The Cramér-Lundberg model with a fluctuating number of clients," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 1-22.
    • Handle: RePEc:eee:insuma:v:112:y:2023:i:c:p:1-22
    DOI: 10.1016/j.insmatheco.2023.05.007
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    References listed on IDEAS

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    1. C. Constantinescu & G. Delsing & M. Mandjes & L. Rojas Nandayapa, 2020. "A ruin model with a resampled environment," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2020(4), pages 323-341, April.
    2. Gabriele Stabile & Giovanni Luca Torrisi, 2010. "Risk Processes with Non-stationary Hawkes Claims Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 415-429, September.
    3. Huang, Gang & Mandjes, Michel & Spreij, Peter, 2016. "Large deviations for Markov-modulated diffusion processes with rapid switching," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1785-1818.
    4. Mandjes, Michel & Mannersalo, Petteri & Norros, Ilkka & van Uitert, Miranda, 2006. "Large deviations of infinite intersections of events in Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1269-1293, September.
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    More about this item

    Keywords

    Cramér-Lundberg; Large deviations; Ruin probability; Exponential tightness;
    All these keywords.

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • P - Political Economy and Comparative Economic Systems

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