Study On The Taxation And Dividend Problems In Actuarial Models

Taxation and dividend are two important problems in actuarial models, and are two hot areas of research in the field of accuracy. Taxation is a problem from the perspective of government, it involves two subjects:One is that the tax rate is given, we study the actuarial variables such as the ruin probability, the discounted tax payments until ruin and the optimal tax barrier and so on; the second is to find an optimal taxation strategy which maximize the expected discounted tax payments until ruin. Dividend is a problem from the perspective of shareholder, it involves two subjects too:One is that the dividend strategy is given, we study the actuarial variables such as the ruin probability, the discounted dividend payments until ruin and the ruin time and so on; the second is to find an optimal dividend strategy which maximize the expected discounted dividend payments until ruin. This paper focus on the taxation and dividend problems in actuarial models, it organized as follows:In chapter 1, we introduce some actuarial models related to this paper. In addi-tion, we summarize the research backgrounds and situations of taxation and dividend problems.In chapter 2, we study the optimal taxation strategy and optimal value function in the classical risk model. By using the theory of stochastic control, the Hamilton-Jacobi-Bellman(HJB) equation associated with the optimal value function is given. By using the theory of viscosity solution, we characterize the optimal value function as the smallest increasing, bounded and Lipschitz viscosity solution of the associated HJB equation, prove the existence of optimal taxation strategy and give the method of constructing it. In addition, the explicit expression of the optimal value function is obtained under the assumption that the claim sizes are distributed with Gamma(2,1).In chapter 3, we consider the taxation problems in the Markov-modulated dual model. Assume that taxes are paid according to a loss-carry-forward system. When no tax is paid, a system of integro-differential equations for the Laplace transform of passage time and ruin time are firstly obtained. For exponentially distributed gains, a system of differential equations for the survival probability, the Laplace transform of the ruin time and the expected discounted tax payments with the corresponding boundary conditions are also derived. In addition, a numerical illustration is presented.In chapter 4, section 4.1 studies the taxation problems in the classical risk model with capital injections. Assume that the taxes are paid according to a loss-carry-forward system. Once the surplus is below 0, we let the surplus attain 0 avoiding ruin by capital injections. Applying the "differential argument", the integro-differential equation for the total expected discounted tax payments minus the total expected discounted costs of capital injections (V(x)) is derived, the limitation of V(x) is given when the initial surplus x tends to infinity, and the explicit expression for V(x) is obtained under the assumption that the claim sizes are exponentially distributed. Section 4.2 studies the taxation problems in the dual model with capital injections. Take the same taxation system and capital injection strategy as section 4.1. The explicit expression for the expected discounted tax payments minus the expected discounted costs of capital in-jections is obtained under the assumption that the gains are exponentially distributed.In chapter 5-7, we assume that dividends can only be paid at some randomized observation times, the waiting times between successive times are independent and dis-tributed with a common exponential distribution, and on each observation, the dividend is paid according to a barrier strategy. In Chapter 5, the surplus process before paying dividend is described by the perturbed classical risk model. The integro-differential equations for the expected discounted dividends until ruin and the Laplace transform of ruin time are derived. When the claim is exponentially distributed, the explicit ex-pressions for the expected discounted dividends until ruin and the Laplace transform of ruin time are also obtained. Finally, the optimal dividend barrier which maximizes the expected discounted dividends until ruin is discussed. In chapter 6, the surplus process before paying dividend is described by the dual model with diffusion. The integro-differential equations for the expected discounted dividends until ruin and the Laplace transform of ruin time are derived. When the gains are exponentially distributed, the explicit expressions for the ruin probability, the expected discounted dividends until ru-in, the Laplace transform of ruin time and the expectation of ruin time are obtained. In chapter 7, the surplus process before paying dividend is described by a Sparre-Andersen model, the interclaim times are Phase-type(n) distributions. The integro-differential equations for the expected discounted dividends until ruin and the Laplace transform of ruin time are derived.In chapter 8, the research works and main innovative points are briefly summa-rized, and some problems which need to be further perfected and studied are pointed.