Lévy insurance risk process with Poissonian taxation

The idea of taxation in risk process was first introduced by Albrecher, H. & Hipp, C. Lundberg's risk process with tax. Blätter der DGVFM 28(1), 13-28, 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 Albrecher, H., Cheung, E.C.K. & Thonhauser, S. Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bulletin 41(2), 645-672, 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.