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.
Highlights
The ruin of an insurance company can be viewed as the event of its aggregate claim amount exceeding for the first time the aggregate premium income, modeled by a non-decreasing deterministic function
The joint distribution of the first crossing time and the overshoot of a Levy process over a fixed boundary in infinite time have been considered by Doney (1991), Kluppelberg et al (2004), Doney and Kyprianou (2006), and Eder and
The first crossing time and the overshoot are interpreted as the ruin time and the deficit at ruin
Summary
The ruin of an insurance company can be viewed as the event of its aggregate claim amount exceeding for the first time the aggregate premium income, modeled by a non-decreasing deterministic function. Ruin is equivalent to first crossing of an upper deterministic boundary by a stochastic process modeling the aggregate claim amount. The joint distribution of the first crossing time and the overshoot of a Levy process over a fixed boundary in infinite time have been considered by Doney (1991), Kluppelberg et al (2004), Doney and Kyprianou (2006), and Eder and Kluppelberg (2009). The first crossing time and the overshoot are interpreted as the ruin time and the deficit at ruin. Ruin time and deficit in a classical infinite time risk model, have been considered jointly through a defective renewal equation in terms of what is called Gerber-
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