Lévy insurance risk process with Poissonian taxation

Abstract

The idea of taxation in risk process was first introduced by Albrecher and Hipp (2007), 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 L'{e}vy insurance risk model under taxation is studied. Motivated by the concept of randomized observations proposed by Albrecher et al. (2011b), 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 (Gerber and Shiu (1998)) and the expected discounted tax payments until ruin are derived. The Cram'{e}r-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.