General tax structures for a L\'evy insurance risk process under the Cram\'er condition

TL;DR

This paper extends classical ruin probability estimates to a Levy insurance risk model with tax, deriving asymptotic distributions and present value calculations of tax payments under Cramér's condition.

Contribution

It introduces a novel analysis of the Levy risk process with tax, providing new asymptotic results and extending classical estimates to this more complex setting.

Findings

Derived an analogue of Cramér's estimate for taxed Levy risk models.

Obtained asymptotic distributions of surplus variables at ruin.

Calculated the expected present value of tax payments at ruin.

Abstract

We investigate the Levy insurance risk model with tax under Cram\'er's condition. A direct analogue of Cram\'er's estimate for the probability of ruin in this model is obtained, together with the asymptotic distribution, conditional on ruin occurring, of several variables of interest related to ruin including the surplus immediately prior to ruin (undershoot) and shortfall at ruin (overshoot). We also compute the present value of all tax paid conditional on ruin occurring. The proof involves first transferring results from the model with no tax to the reflected process, and from there to the model with tax. In doing so we also derive new results for the reflected process.