Applications Of Some Kinds Of Extended Risk Models

Risk theory is the most theoretical part in the actuarial mathematics which orig-inates from the management of insurance companies. The basic model in risk theory is the classical Cramer-Lundberg model, which is introduced by Lundberg [78] in 1903. This model is described by a compound Poisson process with the properties of temporal homogeneity and independent increment. The initial study of risk the-ory mainly focuses on several important actuarial variables, for example, the ruin probability, the time of ruin, the surplus immediately prior to ruin, the deficit at ruin and so on. The applications of stochastic process and stochastic analysis in risk theory, especially the martingale method introduced by Gerber etc,make risk the-ory develop rapidly. Many fruitful achievements in characterizing several actuarial variables in the classical risk model have been made.However, it is assumed that the surplus process has temporal homogeneity and independent increment in the classical risk model. In practice, many scenarios do not have this nature. Therefore, in order to give a reasonably realistic description of the actual behavior of the risk movement, all kinds of extended risk models of the classical risk process are introduced in the field of risk theory. In the following, we will present some extensions.First, some stochastic perturbation process are added to the classical risk model. In reality, the income of an insurance company is not deterministic. There are fluc-tuations in the number of customers. The claim arrival intensity may depend on time. The claim and the premium may increase with inflation. To describe these additional uncertainties, one can add some stochastic perturbation process (such as Brownian motion, Levy process, etc) to the original classical risk process.Secondly, the distribution of the inter-claim times are extended. Andersen [11] first introduced the Sparre Andersen risk model in 1957. In this model, assume that the inter-claim times have an arbitrary distribution, which relaxes the exponential assumption for the distribution of the interarrival times. Ever since then, many scholars have been devoted to studying this model. A number of valuable results have been obtained as long as the claim waiting times are Erlang distributed or phase-type distributed. The reader may refer to Dickson and Hipp [32,33], Cheng and Tang [24], Avram and Usabel [19], Li and Garrido [65,66], Albrecher and Boxma [5], Gerber and Shiu [50], Schmidli [97], Ren [92], Li [62,63], Li and Lu [70] and the references therein.Next, the Markov-modulated risk model is introduced. Since the actual insur-ance policies are not static, insurance policies need to change if the environment, such as weather condition, economical or political environment, etc, changes. For example, in car insurance, weather condition plays a major role in the occurrence of accidents. The claim size distribution and the intensity of the claim arrival pro-cess in different weather conditions will be very different. Therefore the insurance policies of insurance companies, such as premium rate, will be different in differ-ent weather conditions. Then it is highly necessary and practical to introduce the Markov-modulated risk model to describe this circumstances. For the details of the Markov-modulated model, we refer the reader to Reinhard [91], Asmussen. [12,13], Bauerle [20], Wu [112], Miyazawa and Takada [83], Adan and Kulkarni [2], Miyazawa [82], Albrecher and Boxma [5], Jacobsen [56], Lu and Li [74], Ng and Yang [86,87], Lu [77], Lu and Li [75] and so on.Finally, one may take the factors of reinsurance, investment, dividends and tax payments into account. As the development of insurance and finance markets, the study of risk theory obviously unfolds the characteristic of embracing the mathe-matical finance, such as reinsurance, investment, dividends, tax payments, etc. An insurance company receives the premium, but it also will face the risk of paying the claim. Sometimes, ruin will occur when the claim is higher than its surplus. There-fore, the insurance company will take the appropriate measures such as reinsurance, investment in finance market. In addition, for instance, considering the risk, the insurer is relatively more and more concerned about their profits. The total divi-dends before ruin is the most representative variable for measuring the profits. The so-called dividends means that the company gives a, part of surplus to the share- holder or the person who provides the initial surplus. Hence, the amount of the total dividends not only represents the company's benefit, but also provides a symbol of the company's competitiveness. Moreover, the factor of tax payments plays a role that can not be ignored. In view of a number of practical factors, extended risk models of the classical risk process becomes more and more important in insurance and finance.On the basis of these backgrounds and current research, my doctoral disserta-tion is mainly devoted to considering the ruin probability, Gerber-Shiu discounted penalty function, dividends problems and tax payments problem in some extended risk models. The dissertation is organized as follows.A brief overview of the classical risk model and some useful definitions, such as the time of ruin, the ruin probability. Gerber-Shiu function, the expected discounted dividends and the expected discounted tax payments, are given in Chapter 1. Fol-lowing that, is the main body of this dissertation.In Chapter 2, we investigate the perturbed classical surplus model. Since the influence of the diffusion term, it leads to that the ruin time may coincide with a certain moment of a claim, or it may happen between two consecutive claims oc-currences. Moreover, to comprehensively study for the ruin probability, we need to discuss the probability that ruin occurs at each instant of claims, and the proba-bility that ruin occurs between two consecutive claims occurrences, as well as the distribution of the ruin time that lies in between two consecutive claims. We give some explicit expressions of Laplace transforms of these quantities mentioned above. The formulae for these Laplace transforms can be inverted by computer in practice, and they are brief and favorable for calculation. Finally, numerical examples are presented to illustrate our results.In Chapter 3, we study two classes of the Sparre Andersen risk model perturbed by diffusion:one is the inter-claim times with generalized Erlang(n) distribution:the other is the inter-claim times with phase-type distribution. We mainly consider the distribution of maximum surplus prior to ruin and related problems. The maximum surplus before ruin is an important indicator of the assets in insurance institutions. For an insurance company, investigating this maximum can not only grasp its abil-ity of withstanding bankruptcy, but also provide an important basis for carrying out other businesses, such as the dividend and the investment. This quantity has been received remarkable attention by Gerber and Shiu [46], Li [63] and Li and Lu [69]. In this chapter, we derive a integro-differential equation with certain bound-ary conditions, describing the maximum surplus. The solution of above equation can be expressed as a linear combination of particular solutions of the correspond-ing homogeneous integro-differential equation. Using the operator Tr introduce by Dickson and Hipp [33], the divided differences technique and nonnegative real part roots of Lundberg's equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. Furthermore, we also give an explicit expression for the expected discounted dividend payments under a barrier dividend strategy.In Chapter 4, we investigate a Markov-modulated risk model perturbed by dif-fusion with a threshold dividend strategy. Under this strategy, suppose that if the surplus is above certain threshold level before ruin, dividends are continuously paid at a constant rate which does not exceed the premium rate, and if the surplus below the threshold prior to ruin. no dividends are paid. In this model, we deriv some matrix forms of homogeneous integro-diflerential equations satisfied by the Gerber-shiu function and the expected present value of dividends until ruin. Apply-ing the operator Tτand divided differences on matrices, which were extended by Li and Lu [69] and Lu and Li [75], we obtain analytical expressions for these quantities mentioned above. Finally, we give some numerical examples.In Chapter 5, we discuss the classical risk process with investment and tax pay-ments. Albrecher and Hipp [9] extended the study to incorporate tax payments. They proposed a loss-carried-forward tax scheme with a constant tax rateγ∈[0,1). That is, tax is paid at a fixed rateγof the premium income, whenever the surplus is at a running maximum, namely a profitable situation. Recently, tax payments problem has gained much interested in the actuarial literature.In Sections 5.1, we investigate a constant interest risk model with tax payments according to a loss-carried-forward system. We make two aspects of extensions:on the one hand, the taxation is paid on both the premium and the interest income:on the other hand, tax is paid according to a surplus-dependent tax rateγ(x)∈[0,1), which will be a constant not always. In this model, we derive an explicit expression for the expected discounted tax payments. For the tax authority, it may be better to collect tax only when the surplus has exceeded a threshold. The tax authority is interested in choosing a proper threshold of starting taxation to maximize the expected discounted tax payments. For a constant tax rate, we give a sufficient condition under which a unique level for starting taxation exists. Finally, some nu-merical examples are given in the case of exponentially distributed claim sizes.In Section 5.2, we extend the work of above section. Based on the classical risk model, we take the following three factor into account:when the surplus is in a profitable situation, the insurer may pay a certain proportion of the income as tax payments；whenever the surplus is nonnegative, the insurer will invest in a risk-free asset with constant interest force; whenever the surplus is on deficit, the insurer could borrow money at a debit interest to continue his business. Meanwhile, the in-surer will repay the debts from his income. In this model, the negative surplus may return to a positive level expect that the surplus is bellow a certain critical level. In the latter case, we say that absolute ruin occurs. In this section, the expected discounted tax payments prior to absolute ruin is investigated and the explicit ex-pression is obtained.In Chapter 6, we summarize the thesis and point out our future research.