Optimal loss-carry-forward taxation for the Lévy risk model

In the spirit of Albrecher and Hipp (2007), Albrecher et al. (2008b) and Kyprianou and Zhou (2009), we consider the reserve process of an insurance company which is governed by Rtπ=Xt−∫0tγπ(Sσ)dSσ, where X is a spectrally negative Lévy process with the usual exclusion of negative subordinator or deterministic drift, St:=max0≤σ≤tXσ the running supremum of X, and γπ(St) the rate of loss-carry-forward tax at time t which is subject to the taxation allocation policy π and is a function of St. The objective is to find the optimal policy which maximizes the total (discounted) taxation pay-out: Ex∫0τπe−ctγπ(St)dSt, where Ex and τπ refer to the expectation corresponding to the law of X such that X0=x, and the time of ruin, respectively. With the scale function of X denoted by Wc(x) and γπ(⋅) allowed to vary in [α,β](0≤α≤β<1), two situations are considered. (a)∫0∞(Wc(y))1−11−β(Wc)″(y)[(Wc)′(y)]2dy≥0. It is shown that the optimal strategy is to always pay tax at the maximum rate β.(b)∫0∞(Wc(y))1−11−β(Wc)″(y)[(Wc)′(y)]2dy<0. Then the optimal strategy prescribes to pay tax at the smallest rate α when the reserve is below some critical level u0, and to pay at the maximum rate β when the reserve is above u0.