Optimal Loss-carry-forward Taxation For Insurance Risk Models

The problem of optimal loss-carry-forward taxation of insurance company has re-cently been gaining a lot of attention in actuarial science. From certain point view, studies of loss-carry-forward taxation provide theoretical support for designing taxation policies of the government taxation authority. This thesis is focused on the issue of optimal taxation payout. The thesis is organized as follows:In Chapter1, a short review of existing literature concerning the problem of loss-carry-forward taxation is presented.In Chapter2, motivated by Albrecher and Hipp (2007), Albrecher et al.(2008) and Kyprianou and Zhou (2009), we consider the reserve process of an insurance company which is governed by Rtπ=Xt-f0tγπ(Sσ)dSσ, where X is a spectrally negative Levy process with the usual exclusion of negative subordinator or deterministic drift, St=(?) Xσ the running supremum of X, and γπ(St) the rate of loss-carry-forward taxation 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 f0Tπe-etγπ(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 γx(-) allowed to vary in [α,β](0≤α≤β<1), two situations are considered:(a) It is shown that the optimal strategy is to always pay taxation at the maximum rate β.(b) Then the optimal strategy prescribes to pay taxation at the smallest rate a when the reserve is below some critical level u0, and to pay maximum taxation rate β when the reserve is above u0.In Chapter3, we discuss the problem of maximizing the expected cumulated dis-counted loss-carry-forward tax payments until ruin in the Cramer-Lundberg risk model including a constant force of interest. The optimal taxation return function is identi-fied as the classical solution of the associated Hamilton-Jacobi-Bellman equation and the optimal taxation strategy in this risk model with interest is derived, which is of band type. Finally, an example is constructed for exponential claim sizes, m which closed-form expression for the optimal taxation return function is given.