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Ruin and Deficit Under Claim Arrivals with the Order Statistics Property

Dimitrova, D. S. ORCID: 0000-0003-3169-2735, Kaishev, V. K. & Ignatov, Z. G. (2018). Ruin and Deficit Under Claim Arrivals with the Order Statistics Property. Methodology and Computing in Applied Probability, doi: 10.1007/s11009-018-9669-5


We consider an insurance risk model with extended flexibility,
under which claims arrive according to a point process with an order
statistics (OS) property, their amounts may have any joint distri-
bution and the premium income is accumulated following any non-
decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (2000, 2004, 2006) and Lef`evre and Loisel (2009) for the case of stationary Poisson claim arrivals and by Lef`evre and Picard (2011, 2014), for OS claim arrivals.

Publication Type: Article
Additional Information: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Publisher Keywords: Order statistics point process, Appell polynomials, Hessenberg determinants, Risk process, Ruin probability, First crossing time, Overshoot
Subjects: H Social Sciences > HA Statistics
H Social Sciences > HD Industries. Land use. Labor > HD61 Risk Management
Q Science > QA Mathematics
Departments: Bayes Business School > Actuarial Science & Insurance
Text - Accepted Version
Available under License Creative Commons Attribution.

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