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Ruin and deficit under claim arrivals with the order statistics property. (English) Zbl 1427.91078

Summary: 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 the last two authors [Scand. Actuar. J. 2000, No. 1, 46–62 (2000; Zbl 0958.91030); J. Appl. Probab. 41, No. 2, 570–578 (2004; Zbl 1048.60079); ibid. 43, No. 2, 535–551 (2006; Zbl 1108.60080)] and C. Lefèvre and S. Loisel [Methodol. Comput. Appl. Probab. 11, No. 3, 425–441 (2009; Zbl 1170.91414)] for the case of stationary Poisson claim arrivals and by C. Lefèvre and P. Picard [Insur. Math. Econ. 49, No. 3, 512–519 (2011; Zbl 1229.91162)] for OS claim arrivals.

MSC:

91B05 Risk models (general)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G51 Processes with independent increments; Lévy processes
91G05 Actuarial mathematics
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