On The Risk Models With Tax

In this thesis, four generalized Cramer-Lundberg risk models and a dual risk model are studied. The thesis is divided into five chapters.In Chapter 1,we give the overall introduction.In Chapter 2, we consider the generalized Cramer-Lundberg risk model includ-ing tax payments. We investigate how tax payments affect the behavior of a Cramer-Lundberg surplus process by defining an expected discounted penalty function at ruin. An explicit expression for this function is derived by solving a differential equa-tion. Consequently,the exact formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims.In Chapter 3, we generalize the classical Cramer-Lundberg risk model by incor-porating a debit interest and tax payments. When the portfolio is in a profitable situa-tion, the insurer may pay a certain proportion of the premium income as tax payments. When the portfolio is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from his premium income. The negative surplus may return to a positive level except that the surplus is below a certain critical level. When the surplus is below the certain critical level, we say that absolute ruin occurs at this moment. Absolute ruin questions are studied by defining an expected discounted penalty function at absolute ruin. First, we derive a system of integro-differential equations satisfied by the func-tion. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability equals asymptotically to a proportion of the absolute ruin probability for the Cramer-Lundberg surplus process with debit interest only, whether the claims are light-tailed or heavy-tailed.Third, explicit expressions for the expected discounted to-tal sum of tax payments until absolute ruin and the Laplace-Stieltjes transform (LST) of the total duration of negative surplus are derived.In addition, we give explicit expressions for the function for exponential claims.Furthermore, to be even closer to the real situation, in the last four sections of Chapter 3, we modify the afore mentioned generalized Cramer-Lundberg risk model with debit interest and tax payments through inclusion of a credit interest. That is, the insurer shall earn credit interests at a constant rate whenever the surplus is positive. When the claims are long-tailed, we show that the absolute ruin probability is asymp-totically equal to a ratio of the absolute ruin probability of the Cramer-Lundberg risk model with debit interest and credit interest. Subsequently, the asymptotical behavior of the absolute ruin probability for the case of light-tailed claims is discussed.The explicit expressions for the absolute ruin probability for exponential claims are also given.Chapter 4 is dedicated to the study of a generalized Cramer-Lundberg risk model with taxes paid according to a loss-carry forward system and dividends paid under a threshold strategy. The ruin quantities are discussed by defining an expected dis-counted penalty (EDP) function at ruin. First, the analytical solution of an integro-differential equation satisfied by the EDP function is derived. Second, closed-form expressions for the expected discounted total sum of tax payments until ruin and the probability function of the total number of taxation periods over the lifetime of the surplus process are obtained. Third, closed-form expressions for the expected ac-cumulated discounted dividends until ultimate ruin and the expected accumulated discounted dividends paid between two consccutive taxation periods are presented. Additionally, for exponential individual claims, we give explicit expressions for the ultimate ruin probability, the Laplace-stieltjes transform of the time to ultimate ruin, the expected discounted total sum of tax payments and the expected accumulated discounted dividends until ruin. Numerical illustrations of the expected accumulated discounted dividends until ruin and the optimal threshold are also given.The aim of Chapter 5 is to consider the dual risk process in which dividends are paid under a threshold strategy and tax payments are paid according to a loss-carry forward system. For this model, the ruin probability, the expected discounted divi-dends and the expected discounted tax payments are investigated. Integral equations, integro-differential equations and analytical expressions for them are derived.Finally for exponential individual claims, closed-form expressions for the ultimate ruin prob-ability, the expected accumulated discounted tax and the expected discounted divi-dends until ruin are given.