Ruin and Deficit Under Claim Arrivals with the Order Statistics Property

Dimitrina S. Dimitrova (), Zvetan G. Ignatov () and Vladimir K. Kaishev ()
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Dimitrina S. Dimitrova: Cass Business School City, University of London
Zvetan G. Ignatov: Sofia University “St Kliment Ohridski”
Vladimir K. Kaishev: Cass Business School City, University of London

Methodology and Computing in Applied Probability, 2019, vol. 21, issue 2, 511-530

Abstract: Abstract 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 distribution 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 (Scand Actuar J 2000(1):46–62, 2000; J Appl Probab 41(2):570–578, 2004; J Appl Probab 43:535–551, 2006) and Lefèvre and Loisel (Methodol Comput Appl Probab 11(3):425–441, 2009) for the case of stationary Poisson claim arrivals and by Lefèvre and Picard (Insurance Math Econom 49:512–519, 2011; Methodol Comput Appl Probab 16:885–905, 2014), for OS claim arrivals.

Keywords: Order statistics point process; Appell polynomials; Hessenberg determinants; Risk process; Ruin probability; First crossing time; Overshoot; Primary 60K30; Secondary 60K99 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-018-9669-5

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