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Lévy insurance risk process with Poissonian taxation. (English) Zbl 1401.91216
Summary: The idea of taxation in risk process was first introduced by H. Albrecher and C. Hipp [Bl. DGVFM 28, No. 1, 13–28 (2007; Zbl 1119.62103)], who suggested that a certain proportion of the insurer’s income is paid immediately as tax whenever the surplus process is at its running maximum. In this paper, a spectrally negative Lévy insurance risk model under taxation is studied. Motivated by the concept of randomized observations proposed by H. Albrecher et al. [ASTIN Bull. 41, No. 2, 645–672 (2011; Zbl 1239.91072)], we assume that the insurer’s surplus level is only observed at a sequence of Poisson arrival times, at which the event of ruin is checked and tax may be collected from the tax authority. In particular, if the observed (pre-tax) level exceeds the maximum of the previously observed (post-tax) values, then a fraction of the excess will be paid as tax. Analytic expressions for the Gerber-Shiu expected discounted penalty function and the expected discounted tax payments until ruin are derived. The Cramér-Lundberg asymptotic formula is shown to hold true for the Gerber-Shiu function, and it differs from the case without tax by a multiplicative constant. Delayed start of tax payments will be discussed as well. We also take a look at the case where solvency is monitored continuously (while tax is still paid at Poissonian time points), as many of the above results can be derived in a similar manner. Some numerical examples will be given at the end.

MSC:
91B30 Risk theory, insurance (MSC2010)
91B64 Macroeconomic theory (monetary models, models of taxation)
60G51 Processes with independent increments; Lévy processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
60J75 Jump processes (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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