Ruin And Dividend Problems In Insurance Risk Theory

In classical risk theory, one of the mostly concerned problems is to find the ruin prob-ability and other related actuarial quantities. It has been an active area of research from the days of Lundberg all the way up to today. Furthermore, ruin theory has a wide range of applications to other fields of applied probability, such as queueing theory and mathe-matical finance (pricing of barrier options, credit products etc.). So it is widely believed that ruin theory is still important for modern risk theory. Dividend payments, as another important criterion, were first proposed by De Finetti [19]. He suggested to look for the expected discounted sum of dividend payments until the time of ruin in a simple discrete time model, and he found that the optimal dividend strategy must be a barrier strategy. From then on, dividend problems (with a constant barrier) have been studied by many authors under more general and more realistic model assumptions. My doctoral disserta-tion is mainly devoted to studying the ruin and dividend problems in some risk models. It includes two types of problems:one is some optimal stochastic control problems related to ruin and dividend payments in the continuous time models (see Chapters2and3), the other is the ruin and dividend problems in some discrete time models (see Chapter4-6).Dynamic stochastic optimization arises in decision-making problems under uncertainty, and finds a large number of applications in insurance, finance, economics and management. The main goal in a stochastic optimization problem is to find the optimal control (strategy) process and the corresponding optimal objective function. It took some more time until the first papers on the combination of the insurance and stochastic control theory appeared (e.g. Asmussen and Taksar [5], Browne [8]). Since then, there have been a series of papers in which the dynamic programming approach and the Hamilton-Jacobi-Bellman (HJB) equation were used to solve the optimal control problems in insurance. The core problem in this field mainly includes the problem of optimal reinsurance, optimal investment and optimal dividends for insurers. Among them, most are based on the diffusion models and the classical risk models.For the purpose of reducing its risk, the insurer would like to buy some reinsurance. In most of the literature, premium is always assumed to be calculated via the expected value principle for mathematical convenience. However, one can argue that two risks with the same mean may appear very different and the premium of them should also be different. So it may be not reasonable for the expected value principle sometimes. On the other hand, the exponential premium principle which is the so-called zero utility principle, plays an important role in insurance mathematics and actuarial practice. It has many nice properties and is widely used in mathematical finance to price various insurance products in the market, see Musiela and Zariphopoulou [62], Young and Zariphopoulou [85], Young [84] and Moore and Young [61]. So we are interested in some optimal control problems with premium calculated by the exponential premium principle in Chapters2and3. Under the exponential premium principle, the risk control becomes nonlinear which makes the problems more complicated than those under the expected value principle. We choose proportional reinsurance in my dissertation for convenience.In Chapter2, we consider the optimal dividend payments in the framework of diffusion model. The controlled diffusion model is established as an approximation of the classical risk model with proportional reinsurance under exponential premium calculation. Zhou and Yuen [90] studied the analogous problem with the reinsurance premium be calculated via the variance principle. They obtained some results different from those in L(?)kka and Zervos [55] where the reinsurance premium was calculated by the expected value principle. Our problem is a nonlinear stochastic control problem which is more complicated than that in Zhou and Yuen [90]. Besides, we consider both non-cheap and cheap reinsurance instead of only cheap reinsurance considered in Zhou and Yuen [90]. Our objective is to maximize the expected discounted dividends until ruin. Explicit expressions for the value function and the corresponding optimal strategies are obtained in two cases which depend on whether the dividend rates are bounded. In the case of unbounded dividend rates (both non-cheap and cheap reinsurance), we show that the optimal dividend strategy is a barrier strategy and there exists a common switch level for optimal reinsurance and dividend strategies. These are the same as those in Zhou and Yuen [90]. But for the case in which dividend rates are bounded by a positive constant M, our results for non-cheap reinsurance are somewhat different from those in Zhou and Yuen [90]. Zhou and Yuen [90] showed that the optimal dividend policy is always a threshold dividend strategy with a barrier, and there is no need to increase the risk (even when the reserve increases) when the reserve reaches the threshold level of the dividend pay-outs. But that is just the case for large enough M in our paper; see Subsection2.4.1for details. For small M, the dividends should be paid at the maximum rate at all times and the optimal proportional reinsurance is a constant. Finally, we give a numerical example in Section2.5, which illustrates the effects of a (the risk aversion of the reinsurance company) on the optimal value function and retention level for reinsurance. We find that the impact of a for the value function wears off as a increases, and the retention level increases as a increases for small reserve, but it is not the case for large reserve.In Chapter3, we study the optimal investment and proportional reinsurance policies of an insurer whose insurance business follows a diffusion perturbed classical risk process. Usually, the explicit expression of the ruin probability can not be derived for the classical risk model. However, the adjustment coefficient is related to the ultimate probability of ruin by the Cramer-Lundberg asymptotic formula and by Lundberg’s inequality. So we also concentrate our attention on the effect of reinsurance on the adjustment coefficient in the compound poisson risk model perturbed by diffusion. We assume that assets can be invested in a risk-free asset and in a risky one. In addition to investment, we also assume that the insurer can purchase proportional reinsurance to reduce the underlying insurance risk. It is worth to mention that, for maximizing the adjustment coefficient, we do not constrain our strategies in the constant strategy sets which is different from those in many literatures, see Liang and Guo [51], Centeno [10], Hald and Schmidli [32], Centeno and Guerra [11] and Guerra and Centeno [30]. We first study the problem of maximizing the expected exponential utility of terminal wealth, and then employ the ob-tained result to solve the problem of maximizing the adjustment coefficient. In both of the problems, explicit expressions for their optimal value functions and the corresponding op-timal strategies are obtained by solving the corresponding HJB equations. Furthermore, we show that both the maximum adjustment coefficient and its corresponding optimal strategy are strictly monotone functions with respect to a (the risk aversion of the rein-surance company) and β (the uncertainty of the insurance company). We also give an upper bound on the ruin probability in Section3.4. Besides, we shall mention that the method used in Hald and Schmidli [32] is invalid in this paper. However, we can use our method to derive Theorem1in Hald and Schmidli [32].Markov-modulated risk models, where the surplus processes are influenced by an en-vironmental Markov chain, enable us to capture the feature that insurance policies are dependent on the environment, such as weather condition, economical or political envi-ronment etc.. Since it is more realistic than the classical risk model, it becomes more and more popular and has attracted a lot of attention recently.In a Markov-modulated risk model, the premiums, the claims and the claim num- ber process are usually assumed to be (conditional) independent given the environmental Markov chain, that is, they only depend on the current state of the Markov chain. How-ever, this (conditional) independence assumption maybe somewhat too strong in some applications. A semi-Markovian dependence structure, where the claims and the inter-claim times not only depend on the current state but also depend on the next state of the environmental Markov chain, was first studied by Janssen and Reinhard [41]. Reinhard and Snoussi [65,66] considered a discrete time semi-Markov risk model, which assumes individual claims are influenced by a Markov chain with finite state space and there is autocorrelation among consecutive claim sizes. They derived recursive formulas for the distribution of the surplus just prior to ruin and that of the deficit at ruin in a special case, where a strict restriction on the total claim size was imposed. But what will happen if the restriction is released? Without the restriction imposed on the distributions of the claims, the discrete time semi-Markov risk model of study includes several existing risk models such as the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims) as special cases.In Chapter4, we study the problem of expected discounted dividend payments in the discrete time semi-Markov risk model. We first consider the special case described in Reinhard and Snoussi [65,66]. Employing the method of Reinhard and Snoussi [65,66] and taking advantage of the boundary conditions for barrier dividend strategy, we obtain matrix-form expressions of the expected discounted dividends until ruin for the m-state model in Section4.3. Next, we tackle the same problem for general case of the model, i.e., there is not any assumption about the total claim size. Since the method used in Section4.3is invalid for the general case, we develop a new method in Section4.4. Combining the technique of generating functions, the theory of difference equations and the boundary conditions for barrier dividend strategy, we also give the matrix-form expressions of the expected discounted dividends until ruin for two-state and three-state models. The method can be applied to a model involving any finite states, however, such a generalization will make the derivations of the expected discounted dividends tremendously complicated and tedious. At the end of this chapter, a numerical example is presented, which shows that the results obtained through different methods are equivalent. Through this example, we also observe that the optimal dividend barrier b*, which maximizes the expected discounted dividend payments Vi(u, b),i=1,2, is affected by the initial state i, the initial surplus u as well as the discount factor v.In Chapter5, we consider the survival probability for the discrete semi-Markov risk model described in Chapter4. In this chapter, we assume that the positive safety loading condition holds so that ruin is not certain. For the general situation, the approach used in Reinhard and Snoussi [65,66] is not valid any more even for m=2, so a new method should be developed. Employing the technique of generating functions, we derive the recursive formulae for survival probability in the two-state model, where two cases shall be distinguished according to the distributions of the claims. Having obtained the explicit recursive formula for Φi(u), next we need to determine the initial values Φi(0),i=1,2. Without knowing them, one will not be able to apply the obtained results. In Chapter4, we took full advantage of the boundary conditions for barrier dividend strategy to obtain the initial values for the expected discounted dividends. Unfortunately, the method is invalid in this chapter. In order to calculate Φ1(0) and02(0), we shall make an effort to find two equations of them, see Section5.3. On the other hand, since the model of study embraces some existing discrete-time risk models including the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims), the present chapter generalizes the study of ruin probability for these risk models; see Section5.4for details. Finally, we present some numerical examples to illustrate the application of our results.In Chapter6, we propose a discrete-time NCD risk model that incorporates the well-known No Claims Discount (NCD) system (or bonus-malus system (BMS)) in the car insurance industry. Such a system penalizes policyholders at fault in accidents by sur-charges, and rewards claim-free years by discounts. For simplicity, only two levels of premium are considered in the given model and recursive formulae are derived for its ul-timate ruin probabilities. Then the impact of the NCD system.on ruin probabilities is examined through numerical examples. At last, the joint probability of ruin and deficit at ruin is also considered. For discrete risk models with non-integer irregular premiums (or surpluses), how to build a recursive framework for calculation purposes is still an open problem. Although the attempt in this paper does not cover the most general case, it might give readers a hint when searching for a usable solution.At the end of this part, we state some contributions of this dissertation as follows.· The problem studied in Chapter2is a new kind of nonlinear regular-singular stochas-tic control problem. We assume that the reinsurance premium is calculated according to the exponential premium principle. Under the exponential premium principle, the risk control becomes nonlinear which makes the problem more complicated than that under the expected value principle (or variance principle). Our results show some interesting aspects which are different from those papers considering the expected value principle (or variance principle). Besides, we consider both non-cheap and cheap reinsurance instead of only cheap (or non-cheap) reinsurance considered in many papers.·In Chapter3, the optimal investment and reinsurance problem for a jump diffusion model is considered. For maximizing the adjustment coefficient, we do not constrain our strategies in the constant strategy sets which is different from those in many literatures. Besides, our method is new and can be used to derive Theorem1in Hald and Schmidli [32]. However, the method used in Hald and Schmidli [32] is invalid in this paper since the problem we considered is nonlinear.· Ruin and dividend problems in a discrete time semi-Markov risk model are considered in Chapters4and5. Calculations of the classical actuarial quantities for this model are only partly solved under a rather strict restriction, see Reinhard and Snoussi [65,66]. Without the strict restriction, the model of study includes several existing risk models such as the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims) as special cases. By totally new methods, the recursive formulae for survival probability and expected discounted dividends are derived in this dissertation.· In Chapter6, we consider some ruin problems in a discrete time risk model with non-integer irregular premiums (or surpluses). For this kind of model, how to build a usable recursive framework for calculation purposes is still an open problem. Although the attempt in Chapter6does not cover the most general case, it might give readers a hint when searching for a usable solution.