保险风险理论中的破产和分红问题
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摘要
计算破产概率等相关的精算量是经典风险理论中最为关心的问题之一。从Lundberg时期至今,它一直都是一个很活跃的研究领域。此外,破产理论在其他应用概率领域具有广泛的应用,例如排队论和数理金融(障碍期权、信用产品的定价等)。因此,破产理论在现代风险理论中仍然具有非常重要的作用。分红作为另一个重要的准则首先由De Finetti [19]提出。在该文中,他主要考虑一个简单离散模型下的直到破产前的期望折现累计分红量,并发现最优的分红策略是一个边界策略。从此,一大批学者开始研究各种更加一般和更加实际的模型下的(带有一个常数边界的)分红问题。我的博士论文也致力于研究某些风险模型下的破产和分红问题。它主要包含两类问题：一类是连续时间模型下的某些与破产和分红相关的最优随机控制问题(见第2和3章),另一类是某些离散时间模型下的破产和分红问题(见第4-6章)。
动态随机优化起源于具有不确定性的决策问题,它在保险、金融、经济和管理领域具有广泛的应用。随机优化问题的目标一般是为了寻找最优的控制(决策)过程和相应的最优目标函数。保险学和随机控制理论相结合的文献经历了很长一段时间才得以出现,最初的文献为Asmussen and Taksar [5]和Browne [8]。从此以后,有一系列的文献利用动态规划原理和HJB方程的方法来解决保险中的最优控制问题。这个领域的核心问题包括保险公司的最优再保险、最优投资和最优分红问题。其中,大部分都是考虑扩散模型和经典风险模型。
为了降低自身的风险,保险公司通常会购买适量的再保险。为了数学上的方便,大多数文献都假定保费是按照期望值原理来收取的。但是,均值相同的两个风险之间的差异可能很大,那么对它们所收取的保费也应该不同。因此,期望值原理有时未必合理。另一方面,被称为零效用准则的指数保费原理在保险数学和精算实务中都发挥着重要作用。它具有很多好的性质并且被广泛应用于数理金融中的保险产品定价,见Musiela and Zariphopoulou [62], Young and Zariphopoulou [85], Young [84]和Moore and Young[61]。因此,我们也对指数保费原理下的某些最优控制问题感兴趣,见第2和3章。在指数保费原理下,保险公司的风险控制是非线性的,它使得所考虑的问题比期望值原理下的相应问题更复杂。为了简单起见,本文假设保险公司购买的是比例再保险。
在第2章中,我们考虑一个扩散模型下的最优分红问题。该控制的扩散模型是通过对具有比例再保险的经典风险模型扩散逼近得到的,其中再保险的保费是按照指数保费原理来计算的。Zhou and Yuen [90]在方差保费原理下考虑了类似的最优分红问题。他们得到了一些与L(?)kka and Zervos [55](其中再保险保费按照期望值原理来计算)中不一样的结果。我们所考虑的问题是比Zhou and Yuen [90]中更复杂的非线性随机控制问题。此外,Zhou and Yuen [90]中只考虑了便宜再保险,而我们同时考虑了非便宜再保险和便宜再保险两种情形。本章的目标是最大化直到破产前的期望折现分红量。对分红率有界和无界两种情形,我们都得到了值函数和相应的最优策略的解析表达式。对无界分红率情形(非便宜再保险和便宜再保险),最优分红策略是一个边界策略,并且最优再保险和最优分红策略具有相同的阀值。这些结果与Zhou and Yuen [90]中的类似。但是,对分红率为有界(界为M)的情形,本文中非便宜再保险情形下的结果与Zhou and Yuen [90]中有所不同。Zhou and Yuen [90]指出最优分红策略总是门槛分红策略,且当盈余达到该门槛之后保险公司的自留水平保持不变(即使盈余不断增加)。但是在本文中,该情况只对充分大的M才成立,具体的见2.4.1小节。而对比较小的M,本文的结果表明,最优的分红策略是始终按最大分红率进行分红,且最优的再保险比例始终是一个常数。最后,我们在第2.5节给出了一个数值例子,它阐释了α(再保险公司的风险厌恶)对最优值函数和保险公司自留水平的影响。我们从中发现,随着a的增大,它对值函数的影响越来越小；当盈余较小时,自留水平随α的增大而增大,然而当盈余较大时情况却比较复杂。
在第3章中,我们考虑保险公司的最优投资和比例再保险问题,其中保险公司的业务由一带扩散扰动的经典风险过程来刻画。对经典风险模型,破产概率的解析表达式通常无法得到。然而,由Cramer-Lundberg渐进公式和Lundberg不等式知道,破产概率与调节系数密切相关。因此,在带扩散扰动的经典风险模型下,我们也着重考虑再保险对调节系数的影响。我们假设资产可以被投资于一个风险资产和一个无风险资产。除了投资,我们允许保险公司购买适当的再保险以减少自身的风险。值得一提的是,对于最大化调节系数,我们并没有将策略集限制于常数策略类,这与绝大多数文献都不同,例如Liang and Guo [51], Centeno [10], Hald and Schmidli [32], Centeno and Guerra [11]和Guerra and Centeno [30]我们首先研究最大化终端财富指数效用的问题,然后将所得的结果应用于最大化调节系数的问题中。对上述的两个问题,通过解相应的HJB方程,我们都得到了最优值函数和相应最优策略的解析表达式。此外,我们证明了最大的调节系数及其相应的最优策略都是a(再保险公司的风险厌恶)和β(保险公司的不确定因素)的严格单调函数。在第3.4节中,我们还给出了破产概率的一个上界。此外,我们应该注意的是,Hald and Schmidli [32]中的方法对本文不适用。然而,用我们的方法却可以得到Hald and Schmidli [32]中的定理1。
马氏调节风险模型,由于其盈余过程受一环境马氏链的影响,它能更好的捕捉保单依赖于环境的特征,例如受天气、经济和政治等环境的影响。因为它比经典风险模型更加贴近现实,近年来受到越来越多的关注。
在马氏调节风险模型中,保费、索赔额大小和索赔数过程在给定环境马氏链下通常都假定是(条件)独立的,即它们都只依赖于马氏链的当前状态。然而,在某些应用中这种(条件)独立的假设有点太强了。Janssen and Reinhard [41]首先提出了一种半马氏相依结构,其中索赔额大小和索赔时间间隔不仅依赖于环境马氏链的当前状态还依赖于下一步要转移到的状态。接着,Reinhard and Snoussi [65,66]研究了一个离散时间的半马氏风险模型,其中假定索赔额大小之间是自相关的且受一有限状态马氏链的影响。在该文中,他们对索赔额的分布做了一个严格限制,并在此情形下得到了破产前盈余和破产赤字的联合分布的递推公式。然而,如果我们把这个严格限制去掉的话将会出现怎样的情况呢?我们发现,如果对索赔额的分布不加限制,所研究的离散时间半马氏模型包含了多个已有的风险模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型。
在第4章中,我们研究了离散时间半马氏风险模型下的期望折现分红问题。在第4.3节中,我们首先考虑了Reinhard and Snoussi [65,66]中所描述的那种特殊情况。借助Reinhard and Snoussi [65,66]的方法,并充分利用边界分红策略的边界条件,我们得到了m个状态模型下的直到破产前期望折现分红的矩阵形式的表达式。接着,在第4.4节中,我们在一般模型(即索赔额分布不加任何假设)下考虑了同样的问题。由于4.3节中所用的方法不适用于一般情形,我们在第4.4节中采用了一个新的方法。利用生成函数的方法,并结合差分方程理论以及边界分红策略的边界条件,我们在2个和3个状态模型下也都给出了直到破产前期望折现分红的矩阵形式的表达式。我们的方法可以适用于包含任意个状态的模型,然而,这样的推广将会使得推导过程变得极其复杂和冗长。在本章的结尾,我们给出了一个数值例子(满足4.3节中的限制条件),从中证实了利用两种不同方法所得的结果是一致的。通过这个例子,我们也注意到使得期望折现分红量Vi(u,b),i=1,2,达到最大的最优分红边界b*不仅依赖于初始状态i,还依赖于初始盈余u和折现因子v。
在第5章中,我们在第4章所叙述的模型下考虑了保险公司的生存概率。为了保证保险公司不会必然破产,在这一章里我们假设正安全负荷条件成立。即使对m=2的情形,Reinhard and Snoussi [65,66]中所采用的方法对本文也无法适用,所以我们也必须采用一个新的方法。利用生成函数的方法,我们得到了两状态模型下生存概率φi(u)的递推公式,它根据索赔额分布的情况分为两种情形。得到φi(u)的递推公式后,我们还必须确定两个初始值Φi(0),i=1,2.只有这样,我们才能利用之前所得到的递推公式进行计算。在第4章中,我们充分利用了边界策略的边界条件得到了期望折现分红的初始值。然而,这个方法在本章已不再适用。为了得到φ1(0)和φ2(0),我们必须想方设法找到两个与他们相关的方程,见第5.3节。另一方面,由于我们所考虑的模型包含了多个已有的离散时间模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型,本章的结果推广了这些模型下破产概率的研究,具体的见第5.4节。最后,我们给出了几个数值例子来说明我们的结果。
在第6章中,我们研究了一个离散时间的NCD风险模型,它包含了汽车保险业中著名的无赔款优待折扣系统(或奖惩系统)。该系统通过收取附加费惩罚出事故的投保人,而对无索赔的投保人给予优待折扣。为了简单起见,我们假设保费只按两个层次收取,并得到了最终破产概率的递推公式。然后通过几个数值例子来考察NCD系统对破产概率的影响。最后,我们还考虑了最终破产和破产赤字的联合分布。对于具有非整数且不规则保费(或盈余)的离散风险模型,如何建立一个有效的递推公式仍然是尚待解决的问题。本文虽然不是对一般情形进行的尝试,但部分解决了该问题并对一般情形下问题的解决具有启发作用。
作为这部分的结尾,我们介绍一下这篇博士论文的主要贡献,具体如下。
●第2章研究了具有非线性正则-奇异随机控制的最优分红问题。我们假定保费按指数保费原理计算,从而风险控制是非线性的,它比期望值(或方差)保费原理下的相应问题更难以解决。我们的结果展示了一些与期望值保费原理和方差保费原理都不一样的方面。此外,我们同时考虑了便宜和非便宜再保险,而大多数文献都只考虑了其中一种。
●第3章考虑了跳扩散模型下的最优投资和再保险问题。在研究最大化调节系数时,我们并没有将策略集限制于常数策略类,这与绝大多数文献都不同。此外,用我们的方法可以得到Hald and Schmidli [32]中的定理1。然而,Hald and Schmidli[32]中的方法对本文却不适用,因为我们所考虑的问题是非线性的。
·第4和5章考虑了离散半马氏风险模型下的破产和分红问题。该模型的古典精算量的计算问题仅在较强的条件下得到部分解决,见文献Reinhard and Snoussi[65,66]。本文去掉了强加在模型上的限制条件,使得所研究的模型变得非常广泛,它包含了多个已有的风险模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型。本文通过全新的方法,找到了计算分红和破产问题的递推公式。
·第6章研究了具有非整数且不规则保费(或盈余)的离散风险模型下的破产问题。如何在此类模型下建立一个有效的递推公式仍然是尚待解决的问题。本文虽然不是对一般情形进行的尝试,但部分解决了该问题并对一般情形下问题的解决具有启发作用。
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.
动态随机优化起源于具有不确定性的决策问题,它在保险、金融、经济和管理领域具有广泛的应用。随机优化问题的目标一般是为了寻找最优的控制(决策)过程和相应的最优目标函数。保险学和随机控制理论相结合的文献经历了很长一段时间才得以出现,最初的文献为Asmussen and Taksar [5]和Browne [8]。从此以后,有一系列的文献利用动态规划原理和HJB方程的方法来解决保险中的最优控制问题。这个领域的核心问题包括保险公司的最优再保险、最优投资和最优分红问题。其中,大部分都是考虑扩散模型和经典风险模型。
为了降低自身的风险,保险公司通常会购买适量的再保险。为了数学上的方便,大多数文献都假定保费是按照期望值原理来收取的。但是,均值相同的两个风险之间的差异可能很大,那么对它们所收取的保费也应该不同。因此,期望值原理有时未必合理。另一方面,被称为零效用准则的指数保费原理在保险数学和精算实务中都发挥着重要作用。它具有很多好的性质并且被广泛应用于数理金融中的保险产品定价,见Musiela and Zariphopoulou [62], Young and Zariphopoulou [85], Young [84]和Moore and Young[61]。因此,我们也对指数保费原理下的某些最优控制问题感兴趣,见第2和3章。在指数保费原理下,保险公司的风险控制是非线性的,它使得所考虑的问题比期望值原理下的相应问题更复杂。为了简单起见,本文假设保险公司购买的是比例再保险。
在第2章中,我们考虑一个扩散模型下的最优分红问题。该控制的扩散模型是通过对具有比例再保险的经典风险模型扩散逼近得到的,其中再保险的保费是按照指数保费原理来计算的。Zhou and Yuen [90]在方差保费原理下考虑了类似的最优分红问题。他们得到了一些与L(?)kka and Zervos [55](其中再保险保费按照期望值原理来计算)中不一样的结果。我们所考虑的问题是比Zhou and Yuen [90]中更复杂的非线性随机控制问题。此外,Zhou and Yuen [90]中只考虑了便宜再保险,而我们同时考虑了非便宜再保险和便宜再保险两种情形。本章的目标是最大化直到破产前的期望折现分红量。对分红率有界和无界两种情形,我们都得到了值函数和相应的最优策略的解析表达式。对无界分红率情形(非便宜再保险和便宜再保险),最优分红策略是一个边界策略,并且最优再保险和最优分红策略具有相同的阀值。这些结果与Zhou and Yuen [90]中的类似。但是,对分红率为有界(界为M)的情形,本文中非便宜再保险情形下的结果与Zhou and Yuen [90]中有所不同。Zhou and Yuen [90]指出最优分红策略总是门槛分红策略,且当盈余达到该门槛之后保险公司的自留水平保持不变(即使盈余不断增加)。但是在本文中,该情况只对充分大的M才成立,具体的见2.4.1小节。而对比较小的M,本文的结果表明,最优的分红策略是始终按最大分红率进行分红,且最优的再保险比例始终是一个常数。最后,我们在第2.5节给出了一个数值例子,它阐释了α(再保险公司的风险厌恶)对最优值函数和保险公司自留水平的影响。我们从中发现,随着a的增大,它对值函数的影响越来越小；当盈余较小时,自留水平随α的增大而增大,然而当盈余较大时情况却比较复杂。
在第3章中,我们考虑保险公司的最优投资和比例再保险问题,其中保险公司的业务由一带扩散扰动的经典风险过程来刻画。对经典风险模型,破产概率的解析表达式通常无法得到。然而,由Cramer-Lundberg渐进公式和Lundberg不等式知道,破产概率与调节系数密切相关。因此,在带扩散扰动的经典风险模型下,我们也着重考虑再保险对调节系数的影响。我们假设资产可以被投资于一个风险资产和一个无风险资产。除了投资,我们允许保险公司购买适当的再保险以减少自身的风险。值得一提的是,对于最大化调节系数,我们并没有将策略集限制于常数策略类,这与绝大多数文献都不同,例如Liang and Guo [51], Centeno [10], Hald and Schmidli [32], Centeno and Guerra [11]和Guerra and Centeno [30]我们首先研究最大化终端财富指数效用的问题,然后将所得的结果应用于最大化调节系数的问题中。对上述的两个问题,通过解相应的HJB方程,我们都得到了最优值函数和相应最优策略的解析表达式。此外,我们证明了最大的调节系数及其相应的最优策略都是a(再保险公司的风险厌恶)和β(保险公司的不确定因素)的严格单调函数。在第3.4节中,我们还给出了破产概率的一个上界。此外,我们应该注意的是,Hald and Schmidli [32]中的方法对本文不适用。然而,用我们的方法却可以得到Hald and Schmidli [32]中的定理1。
马氏调节风险模型,由于其盈余过程受一环境马氏链的影响,它能更好的捕捉保单依赖于环境的特征,例如受天气、经济和政治等环境的影响。因为它比经典风险模型更加贴近现实,近年来受到越来越多的关注。
在马氏调节风险模型中,保费、索赔额大小和索赔数过程在给定环境马氏链下通常都假定是(条件)独立的,即它们都只依赖于马氏链的当前状态。然而,在某些应用中这种(条件)独立的假设有点太强了。Janssen and Reinhard [41]首先提出了一种半马氏相依结构,其中索赔额大小和索赔时间间隔不仅依赖于环境马氏链的当前状态还依赖于下一步要转移到的状态。接着,Reinhard and Snoussi [65,66]研究了一个离散时间的半马氏风险模型,其中假定索赔额大小之间是自相关的且受一有限状态马氏链的影响。在该文中,他们对索赔额的分布做了一个严格限制,并在此情形下得到了破产前盈余和破产赤字的联合分布的递推公式。然而,如果我们把这个严格限制去掉的话将会出现怎样的情况呢?我们发现,如果对索赔额的分布不加限制,所研究的离散时间半马氏模型包含了多个已有的风险模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型。
在第4章中,我们研究了离散时间半马氏风险模型下的期望折现分红问题。在第4.3节中,我们首先考虑了Reinhard and Snoussi [65,66]中所描述的那种特殊情况。借助Reinhard and Snoussi [65,66]的方法,并充分利用边界分红策略的边界条件,我们得到了m个状态模型下的直到破产前期望折现分红的矩阵形式的表达式。接着,在第4.4节中,我们在一般模型(即索赔额分布不加任何假设)下考虑了同样的问题。由于4.3节中所用的方法不适用于一般情形,我们在第4.4节中采用了一个新的方法。利用生成函数的方法,并结合差分方程理论以及边界分红策略的边界条件,我们在2个和3个状态模型下也都给出了直到破产前期望折现分红的矩阵形式的表达式。我们的方法可以适用于包含任意个状态的模型,然而,这样的推广将会使得推导过程变得极其复杂和冗长。在本章的结尾,我们给出了一个数值例子(满足4.3节中的限制条件),从中证实了利用两种不同方法所得的结果是一致的。通过这个例子,我们也注意到使得期望折现分红量Vi(u,b),i=1,2,达到最大的最优分红边界b*不仅依赖于初始状态i,还依赖于初始盈余u和折现因子v。
在第5章中,我们在第4章所叙述的模型下考虑了保险公司的生存概率。为了保证保险公司不会必然破产,在这一章里我们假设正安全负荷条件成立。即使对m=2的情形,Reinhard and Snoussi [65,66]中所采用的方法对本文也无法适用,所以我们也必须采用一个新的方法。利用生成函数的方法,我们得到了两状态模型下生存概率φi(u)的递推公式,它根据索赔额分布的情况分为两种情形。得到φi(u)的递推公式后,我们还必须确定两个初始值Φi(0),i=1,2.只有这样,我们才能利用之前所得到的递推公式进行计算。在第4章中,我们充分利用了边界策略的边界条件得到了期望折现分红的初始值。然而,这个方法在本章已不再适用。为了得到φ1(0)和φ2(0),我们必须想方设法找到两个与他们相关的方程,见第5.3节。另一方面,由于我们所考虑的模型包含了多个已有的离散时间模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型,本章的结果推广了这些模型下破产概率的研究,具体的见第5.4节。最后,我们给出了几个数值例子来说明我们的结果。
在第6章中,我们研究了一个离散时间的NCD风险模型,它包含了汽车保险业中著名的无赔款优待折扣系统(或奖惩系统)。该系统通过收取附加费惩罚出事故的投保人,而对无索赔的投保人给予优待折扣。为了简单起见,我们假设保费只按两个层次收取,并得到了最终破产概率的递推公式。然后通过几个数值例子来考察NCD系统对破产概率的影响。最后,我们还考虑了最终破产和破产赤字的联合分布。对于具有非整数且不规则保费(或盈余)的离散风险模型,如何建立一个有效的递推公式仍然是尚待解决的问题。本文虽然不是对一般情形进行的尝试,但部分解决了该问题并对一般情形下问题的解决具有启发作用。
作为这部分的结尾,我们介绍一下这篇博士论文的主要贡献,具体如下。
●第2章研究了具有非线性正则-奇异随机控制的最优分红问题。我们假定保费按指数保费原理计算,从而风险控制是非线性的,它比期望值(或方差)保费原理下的相应问题更难以解决。我们的结果展示了一些与期望值保费原理和方差保费原理都不一样的方面。此外,我们同时考虑了便宜和非便宜再保险,而大多数文献都只考虑了其中一种。
●第3章考虑了跳扩散模型下的最优投资和再保险问题。在研究最大化调节系数时,我们并没有将策略集限制于常数策略类,这与绝大多数文献都不同。此外,用我们的方法可以得到Hald and Schmidli [32]中的定理1。然而,Hald and Schmidli[32]中的方法对本文却不适用,因为我们所考虑的问题是非线性的。
·第4和5章考虑了离散半马氏风险模型下的破产和分红问题。该模型的古典精算量的计算问题仅在较强的条件下得到部分解决,见文献Reinhard and Snoussi[65,66]。本文去掉了强加在模型上的限制条件,使得所研究的模型变得非常广泛,它包含了多个已有的风险模型,如(带时间相依索赔的)复合二项模型和(带时间相依索赔的)复合马氏二项模型。本文通过全新的方法,找到了计算分红和破产问题的递推公式。
·第6章研究了具有非整数且不规则保费(或盈余)的离散风险模型下的破产问题。如何在此类模型下建立一个有效的递推公式仍然是尚待解决的问题。本文虽然不是对一般情形进行的尝试,但部分解决了该问题并对一般情形下问题的解决具有启发作用。
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.
引文
[1]Albrecher, H., Boxma,O. J.,2005. On the discounted penalty function in a markov-dependent risk model. Insurance:Mathematics and Economics 37 (3),650-672.
[2]Albrecher, H., Claramunt, M. M., Marmol, M.,2005. On the distribution of dividend payments in a sparre andersen model with generalized erlang(n) interclaim times. Insurance:Mathematics and Economics 37 (2),324-334.
[3]Asmussen, S., Albrecher, H.,2010. Ruin probabilities. Vol.14. World Scientific.
[4]Asmussen, S., H(?)jgaard, B., Taksar, M.,2000. Optimal risk control and dividend dis-tribution policies. example of excess-of loss reinsurance for an insurance corporation. Finance and Stochastics 4 (3),299-324.
[5]Asmussen, S., Taksar, M.,1997. Controlled diffusion models for optimal dividend pay-out. Insurance:Mathematics and Economics 20 (1),1-15.
[6]Bai, L., Guo, J.,2008. Optimal proportional reinsurance and investment with multi-ple risky assets and no-shorting constraint. Insurance:Mathematics and Economics 42 (3),968-975.
[7]Bai, L., Guo, J., Zhang, H.,2010. Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes. Quantitative Finance 10 (10),1163-1172.
[8]Browne, S.,1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research 20 (4),937-958.
[9]Cao, Y., Wan, N.,2009. Optimal proportional reinsurance and investment based on hamilton-jacobi-bellman equation. Insurance:Mathematics and Economics 45 (2), 157-162.
[10]Centeno, M. L.,2002. Measuring the effects of reinsurance by the adjustment coeffi-cient in the sparre anderson model. Insurance:Mathematics and Economics 30 (1), 37-49.
[11]Centeno, M. L., Guerra, M.,2010. The optimal reinsurance strategy-the individual claim case. Insurance:Mathematics and Economics 46 (3),450-460.
[12]Cheng, S,, Gerber, H. U., Shiu, E. S.,2000. Discounted probabilities and ruin theory in the compound binomial model. Insurance:Mathematics and Economics 26 (2), 239-250.
[13]Cheung, E. C., Landriault, D.,2009. Analysis of a generalized penalty function in a semi-markovian risk model. North American Actuarial Journal 13 (4),497-513.
[14]Choulli, T., Taksar, M., Zhou, X.,2001. Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction. Quantitative Finance 1 (6),573-596.
[15]Choulli, T., Taksar, M., Zhou, X.,2003. A diffusion model for optimal dividend distribution for a company with constraints on risk control. SLAM Journal on Control and Optimization 41 (6),1946-1979.
[16]Claramunt, M. M., Marmol, M., Alegre, A.,2003. A note on the expected present value of dividends with a constant barrier in the discrete time model. Bulletin of the Swiss Association of Actuaries 2,149-159.
[17]Cossette, H., Landriault, D., Marceau, E.,2003. Ruin probabilities in the compound markov binomial model. Scandinavian Actuarial Journal 2003 (4),301-323.
[18]Cossette, H., Landriault, D., Marceau, E.,2004. Exact expressions and upper bound for ruin probabilities in the compound markov binomial model. Insurance:Mathe-matics and Economics 34 (3),449-466.
[19]De Finetti, B.,1957. Su un'impostazione alternativa della teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries 2 (1),433-443.
[20]Denuit, M., Marechal, X., Pitrebois, S., Walhin, J. F.,2007. Actuarial modelling of claim counts:Risk classification, credibility and bonus-malus systems. Wiley, com.
[21]Dickson, D. C.,1994. Some comments on the compound binomial model. ASTIN Bulletin 24 (1),33-45.
[22]Dickson, D. C., Waters, H.,2004. Some optimal dividends problems. ASTIN Bulletin 34 (1),49-74.
[23]Diko, P., Usabel, M.,2011. A numerical method for the expected penalty-reward function in a markov-modulated jump-diffusion process. Insurance:Mathematics and Economics 49 (1),126-131.
[24]Dufresne, F.,1988. Distributions stationnaires d'un systeme bonus-malus et proba-bilite de ruine. ASTIN Bulletin 18 (1),31-46.
[25]Fleming, W. H., Soner, H. M.,2006. Controlled Markov processes and viscosity solu-tions. Vol.25. Springer Verlag.
[26]Frangos, N. E., Vrontos, S. D.,2001. Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin 31 (1),1-22.
[27]Gaier, J., Grandits, P., Schachermayer, W.,2003. Asymptotic ruin probabilities and optimal investment. The Annals of Applied Probability 13 (3),1054-1076.
[28]Gerber, H. U.,1988. Mathematical fun with the compound binomial process. ASTIN Bulletin 18 (2),161-168.
[29]Gerber, H. U., Shiu, E. S., Yang, H.,2010. An elementary approach to discrete models of dividend strategies. Insurance:Mathematics and Economics 46 (1),109-116.
[30]Guerra, M., Centeno, M. L.,2008. Optimal reinsurance policy:The adjustment co-efficient and the expected utility criteria. Insurance:Mathematics and Economics 42 (2),529-539.
[31]Guo, X., Liu, J., Zhou, X. Y.,2004. A constrained non-linear regular-singular stochas-tic control problem, with applications. Stochastic processes and their applications 109 (2),167-187.
[32]Hald, M., Schmidli, H.,2004. On the maximisation of the adjustment coefficient under proportional reinsurance. Astin Bulletin 34 (1),75-84.
[33]He, L., Liang, Z.,2008. Optimal financing and dividend control of the insurance com-pany with proportional reinsurance policy. Insurance:Mathematics and Economics 42 (3),976-983.
[34]He, L., Liang, Z.,2009. Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance:Mathematics and Economics 44 (1),88-94.
[35]Hipp, C., Plum, M.,2000. Optimal investment for insurers. Insurance:Mathematics and Economics 27 (2),215-228.
[36]Hipp, C., Plum, M.,2003. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics 7 (3),299-321.
[37]H(?)jgaard, B., Taksar, M.,1998. Optimal proportional reinsurance policies for diffu-sion models. Scandinavian Actuarial Journal 1998 (2),166-180.
[38]H(?)jgaard, B., Taksar, M.,1998. Optimal proportional reinsurance policies for diffu-sion models with transaction costs. Insurance:Mathematics and Economics 22 (1), 41-51.
[39]H(?)jgaard, B., Taksar, M.,1999. Controlling risk exposure and dividends payout schemes:insurance company example. Mathematical Finance 9 (2),153-182.
[40]Irgens, C., Paulsen, J.,2004. Optimal control of risk exposure, reinsurance and in-vestments for insurance portfolios. Insurance:Mathematics and Economics 35 (1), 21-51.
[41]Janssen, J., Reinhard, J. M.,1985. Probabilites de ruine pour une classe de modeles de risque semi-markoviens. ASTIN Bulletin 15 (2),123-134.
[42]Jeanblanc-Picque, M., Shiryaev, A. N.,1995. Optimization of the flow of dividends. Russian Mathematical Surveys 50,257-277.
[43]Kaluszka, M.,2001. Optimal reinsurance under mean-variance premium principles. Insurance:Mathematics and Economics 28 (1),61-67.
[44]Kaluszka, M.,2005. Optimal reinsurance under convex principles of premium calcu-lation. Insurance:Mathematics and Economics 36 (3),375-398.
[45]Landriault, D.,2008. Constant dividend barrier in a risk model with interclaim-dependent claim sizes. Insurance:Mathematics and Economics 42 (1),31-38.
[46]Lemaire, J.,1995. Bonus-malus systems in automobile insurance. Vol.19. Springer.
[47]Li, S.,2006. The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion. Scandinavian Actuarial Journal 2006 (2),73-85.
[48]Li, S., Garrido, J.,2004. On a class of renewal risk models with a constant dividend barrier. Insurance:Mathematics and Economics 35 (3),691-701.
[49]Li, S., Lu, Y.,2009. The distribution of total dividend payments in a sparre andersen model. Statistics & Probability Letters 79 (9),1246-1251.
[50]Liang, Z., Bai, L., Guo, J.,2011. Optimal investment and proportional reinsurance with constrained control variables. Optimal Control Applications and Methods 32 (5), 587-608.
[51]Liang, Z., Guo, J.,2007. Optimal proportional reinsurance and ruin probability. Stochastic Models 23 (2),333-350.
[52]Liang, Z., Yuen, K. C., Guo, J.,2011. Optimal proportional reinsurance and invest-ment in a stock market with ornstein-uhlenbeck process. Insurance:Mathematics and Economics 49 (2),207-215.
[53]Lions, P. L., Sznitman, A. S.,1984. Stochastic differential equations with reflecting boundary conditions. Communications on Pure and Applied Mathematics 37 (4), 511-537.
[54]Liu, C. S., Yang, H.,2004. Optimal investment for an insurer to minimize its proba-bility of ruin. North American Actuarial Journal 8 (2),11-31.
[55]L(?)kka, A., Zervos, M.,2008. Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance:Mathematics and Economics 42 (3), 954-961.
[56]Lu, Y.,2006. On the severity of ruin in a markov-modulated risk model. Scandinavian Actuarial Journal 2006 (4),183-202.
[57]Lu, Y., Li, S.,2005. On the probability of ruin in a markov-modulated risk model. Insurance:Mathematics and Economics 37 (3),522-532.
[58]Lu, Y., Li, S.,2009. The markovian regime-switching risk model with a threshold dividend strategy. Insurance:Mathematics and Economics 44 (2),296-303.
[59]Meng, H., Siu, T.,2011. Optimal mixed impulse-equity insurance control problem with reinsurance. SIAM Journal on Control and Optimization 49 (1),254-279.
[60]Meng, H., Siu, T. K.,2011. On optimal reinsurance, dividend and reinvestment strate-gies. Economic Modelling 28 (1),211-218.
[61]Moore, K. S., Young, V. R.,2003. Pricing equity-linked pure endowments via the principle of equivalent utility. Insurance:Mathematics and Economics 33 (3),497-516.
[62]Musiela, M., Zariphopoulou, T.,2004. An example of indifference prices under expo-nential preferences. Finance and Stochastics 8 (2),229-239.
[63]Ng, A. C, Yang, H.,2006. On the joint distribution of surplus before and after ruin under a markovian regime switching model. Stochastic Processes and Their Applica-tions 116 (2),244-266.
[64]Peng, X., Chen, M., Guo, J.,2012. Optimal dividend and equity issuance problem with proportional and fixed transaction costs. Insurance:Mathematics and Economics 51 (3),576-585.
[65]Reinhard, J., Snoussi, M.,2001. On the distribution of the surplus prior to ruin in a discrete semi-markov risk model. ASTIN Bulletin 31 (2),255-274.
[66]Reinhard, J., Snoussi, M.,2002. The severity of ruin in a discrete semi-markov risk model. Stochastic Models 18 (1),85-107.
[67]Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.,2009. Stochastic processes for insurance and finance. Vol.505. Wiley, com.
[68]Schmidli, H.,2001. Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal 2001 (1),55-68.
[69]Schmidli, H.,2002. Asyrnptotics of ruin probabilities for risk processes under optimal reinsurance policies:the small claim case. Working paper 180, Laboratory of Actuarial Mathematics, University of Copenhagen.
[70]Schmidli, H.,2002. On minimizing the ruin probability by investment and reinsurance. The Annals of Applied Probability 12 (3),890-907.
[71]Shiu, E. S.,1989. The probability of eventual ruin in the compound binomial model. ASTIN Bulletin 19 (2),179-190.
[72]Sotomayor, L. R., Cadenillas, A.,2011. Classical and singular stochastic control for the optimal dividend policy when there is regime switching. Insurance:Mathematics and Economics 48 (3),344-354.
[73]Taksar, M., Markussen, C.,2003. Optimal dynamic reinsurance policies for large insurance portfolios. Finance and Stochastics 7 (1),97-121.
[74]Taksar, M., Zhou, X.,1998. Optimal risk and dividend control for a company with a debt liability. Insurance:Mathematics and Economics 22 (1),105-122.
[75]Tan, J., Yang, X.,2006. The compound binomial model with randomized decisions on paying dividends. Insurance:Mathematics and Economics 39 (1),1-18.
[76]Tremblay, L.,1992. Using the poisson inverse gaussian in bonus-malus systems. ASTIN Bulletin 22 (1),97-106.
[77]Wagner, C.,2001. A note on ruin in a two state markov model. ASTIN Bulletin 31 (2),349-358.
[78]Wagner, C.,2002. Time in the red in a two state markov model. Insurance:Mathe-matics and Economics 31 (3),365-372.
[79]Wei, J., Yang, H., Wang, R.,2010. Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. Journal of Optimization Theory and Applications 147 (2),358-377.
[80]Willmot, G. E.,1993. Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics 12 (2),133-142.
[81]Xiao, Y., Guo, J.,2007. The compound binomial risk model with time-correlated claims. Insurance:Mathematics and Economics 41 (1),124-133.
[82]Yang, H., Zhang, L.,2005. Optimal investment for insurer with jump-diffusion risk process. Insurance:Mathematics and Economics 37 (3),615-634.
[83]Young, V. R.,1999. Optimal insurance under wang's premium principle. Insurance: Mathematics and Economics 25 (2),109-122.
[84]Young, V. R.,2003. Equity-indexed life insurance:pricing and reserving using the principle of equivalent utility. North American Actuarial Journal 7 (1),68-86.
[85]Young, V. R., Zariphopoulou, T.,2002. Pricing dynamic insurance risks using the principle of equivalent utility. Scandinavian Actuarial Journal 2002 (4),246-279.
[86]Yuen, K. C., Guo, J.,2001. Ruin probabilities for time-correlated claims in the com-pound binomial model. Insurance:Mathematics and Economics 29 (1),47-57.
[87]Yuen, K. C., Guo, J.,2006. Some results on the compound markov binomial model. Scandinavian Actuarial Journal 2006 (3),129-140.
[88]Zhang, X.,2008. On the ruin problem in a markov-modulated risk model. Methodol-ogy and Computing in Applied Probability 10 (2),225-238.
[89]Zhang, X. L., Zhang, K. C., Yu, X. J.,2009. Optimal proportional reinsurance and investment with transaction costs, i:Maximizing the terminal wealth. Insurance: Mathematics and Economics 44 (3),473-478.
[90]Zhou, M., Yuen, K. C.,2012. Optimal reinsurance and dividend for a diffusion model with capital injection:Variance premium principle. Economic Modelling 29 (2),198-207.
[91]Zhou, X.,2005. On a classical risk model with a constant dividend barrier. North American Actuarial Journal 9 (4),95-108.
[92]Zhu, J., Yang, H.,2008. Ruin theory for a markov regime-switching model under a threshold dividend strategy. Insurance:Mathematics and economics 42 (1),311-318.
[93]Zhu, J., Yang, H.,2009. On differentiability of ruin functions under markov-modulated models. Stochastic Processes and their Applications 119 (5),1673-1695.
[2]Albrecher, H., Claramunt, M. M., Marmol, M.,2005. On the distribution of dividend payments in a sparre andersen model with generalized erlang(n) interclaim times. Insurance:Mathematics and Economics 37 (2),324-334.
[3]Asmussen, S., Albrecher, H.,2010. Ruin probabilities. Vol.14. World Scientific.
[4]Asmussen, S., H(?)jgaard, B., Taksar, M.,2000. Optimal risk control and dividend dis-tribution policies. example of excess-of loss reinsurance for an insurance corporation. Finance and Stochastics 4 (3),299-324.
[5]Asmussen, S., Taksar, M.,1997. Controlled diffusion models for optimal dividend pay-out. Insurance:Mathematics and Economics 20 (1),1-15.
[6]Bai, L., Guo, J.,2008. Optimal proportional reinsurance and investment with multi-ple risky assets and no-shorting constraint. Insurance:Mathematics and Economics 42 (3),968-975.
[7]Bai, L., Guo, J., Zhang, H.,2010. Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes. Quantitative Finance 10 (10),1163-1172.
[8]Browne, S.,1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research 20 (4),937-958.
[9]Cao, Y., Wan, N.,2009. Optimal proportional reinsurance and investment based on hamilton-jacobi-bellman equation. Insurance:Mathematics and Economics 45 (2), 157-162.
[10]Centeno, M. L.,2002. Measuring the effects of reinsurance by the adjustment coeffi-cient in the sparre anderson model. Insurance:Mathematics and Economics 30 (1), 37-49.
[11]Centeno, M. L., Guerra, M.,2010. The optimal reinsurance strategy-the individual claim case. Insurance:Mathematics and Economics 46 (3),450-460.
[12]Cheng, S,, Gerber, H. U., Shiu, E. S.,2000. Discounted probabilities and ruin theory in the compound binomial model. Insurance:Mathematics and Economics 26 (2), 239-250.
[13]Cheung, E. C., Landriault, D.,2009. Analysis of a generalized penalty function in a semi-markovian risk model. North American Actuarial Journal 13 (4),497-513.
[14]Choulli, T., Taksar, M., Zhou, X.,2001. Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction. Quantitative Finance 1 (6),573-596.
[15]Choulli, T., Taksar, M., Zhou, X.,2003. A diffusion model for optimal dividend distribution for a company with constraints on risk control. SLAM Journal on Control and Optimization 41 (6),1946-1979.
[16]Claramunt, M. M., Marmol, M., Alegre, A.,2003. A note on the expected present value of dividends with a constant barrier in the discrete time model. Bulletin of the Swiss Association of Actuaries 2,149-159.
[17]Cossette, H., Landriault, D., Marceau, E.,2003. Ruin probabilities in the compound markov binomial model. Scandinavian Actuarial Journal 2003 (4),301-323.
[18]Cossette, H., Landriault, D., Marceau, E.,2004. Exact expressions and upper bound for ruin probabilities in the compound markov binomial model. Insurance:Mathe-matics and Economics 34 (3),449-466.
[19]De Finetti, B.,1957. Su un'impostazione alternativa della teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries 2 (1),433-443.
[20]Denuit, M., Marechal, X., Pitrebois, S., Walhin, J. F.,2007. Actuarial modelling of claim counts:Risk classification, credibility and bonus-malus systems. Wiley, com.
[21]Dickson, D. C.,1994. Some comments on the compound binomial model. ASTIN Bulletin 24 (1),33-45.
[22]Dickson, D. C., Waters, H.,2004. Some optimal dividends problems. ASTIN Bulletin 34 (1),49-74.
[23]Diko, P., Usabel, M.,2011. A numerical method for the expected penalty-reward function in a markov-modulated jump-diffusion process. Insurance:Mathematics and Economics 49 (1),126-131.
[24]Dufresne, F.,1988. Distributions stationnaires d'un systeme bonus-malus et proba-bilite de ruine. ASTIN Bulletin 18 (1),31-46.
[25]Fleming, W. H., Soner, H. M.,2006. Controlled Markov processes and viscosity solu-tions. Vol.25. Springer Verlag.
[26]Frangos, N. E., Vrontos, S. D.,2001. Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin 31 (1),1-22.
[27]Gaier, J., Grandits, P., Schachermayer, W.,2003. Asymptotic ruin probabilities and optimal investment. The Annals of Applied Probability 13 (3),1054-1076.
[28]Gerber, H. U.,1988. Mathematical fun with the compound binomial process. ASTIN Bulletin 18 (2),161-168.
[29]Gerber, H. U., Shiu, E. S., Yang, H.,2010. An elementary approach to discrete models of dividend strategies. Insurance:Mathematics and Economics 46 (1),109-116.
[30]Guerra, M., Centeno, M. L.,2008. Optimal reinsurance policy:The adjustment co-efficient and the expected utility criteria. Insurance:Mathematics and Economics 42 (2),529-539.
[31]Guo, X., Liu, J., Zhou, X. Y.,2004. A constrained non-linear regular-singular stochas-tic control problem, with applications. Stochastic processes and their applications 109 (2),167-187.
[32]Hald, M., Schmidli, H.,2004. On the maximisation of the adjustment coefficient under proportional reinsurance. Astin Bulletin 34 (1),75-84.
[33]He, L., Liang, Z.,2008. Optimal financing and dividend control of the insurance com-pany with proportional reinsurance policy. Insurance:Mathematics and Economics 42 (3),976-983.
[34]He, L., Liang, Z.,2009. Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insurance:Mathematics and Economics 44 (1),88-94.
[35]Hipp, C., Plum, M.,2000. Optimal investment for insurers. Insurance:Mathematics and Economics 27 (2),215-228.
[36]Hipp, C., Plum, M.,2003. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics 7 (3),299-321.
[37]H(?)jgaard, B., Taksar, M.,1998. Optimal proportional reinsurance policies for diffu-sion models. Scandinavian Actuarial Journal 1998 (2),166-180.
[38]H(?)jgaard, B., Taksar, M.,1998. Optimal proportional reinsurance policies for diffu-sion models with transaction costs. Insurance:Mathematics and Economics 22 (1), 41-51.
[39]H(?)jgaard, B., Taksar, M.,1999. Controlling risk exposure and dividends payout schemes:insurance company example. Mathematical Finance 9 (2),153-182.
[40]Irgens, C., Paulsen, J.,2004. Optimal control of risk exposure, reinsurance and in-vestments for insurance portfolios. Insurance:Mathematics and Economics 35 (1), 21-51.
[41]Janssen, J., Reinhard, J. M.,1985. Probabilites de ruine pour une classe de modeles de risque semi-markoviens. ASTIN Bulletin 15 (2),123-134.
[42]Jeanblanc-Picque, M., Shiryaev, A. N.,1995. Optimization of the flow of dividends. Russian Mathematical Surveys 50,257-277.
[43]Kaluszka, M.,2001. Optimal reinsurance under mean-variance premium principles. Insurance:Mathematics and Economics 28 (1),61-67.
[44]Kaluszka, M.,2005. Optimal reinsurance under convex principles of premium calcu-lation. Insurance:Mathematics and Economics 36 (3),375-398.
[45]Landriault, D.,2008. Constant dividend barrier in a risk model with interclaim-dependent claim sizes. Insurance:Mathematics and Economics 42 (1),31-38.
[46]Lemaire, J.,1995. Bonus-malus systems in automobile insurance. Vol.19. Springer.
[47]Li, S.,2006. The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion. Scandinavian Actuarial Journal 2006 (2),73-85.
[48]Li, S., Garrido, J.,2004. On a class of renewal risk models with a constant dividend barrier. Insurance:Mathematics and Economics 35 (3),691-701.
[49]Li, S., Lu, Y.,2009. The distribution of total dividend payments in a sparre andersen model. Statistics & Probability Letters 79 (9),1246-1251.
[50]Liang, Z., Bai, L., Guo, J.,2011. Optimal investment and proportional reinsurance with constrained control variables. Optimal Control Applications and Methods 32 (5), 587-608.
[51]Liang, Z., Guo, J.,2007. Optimal proportional reinsurance and ruin probability. Stochastic Models 23 (2),333-350.
[52]Liang, Z., Yuen, K. C., Guo, J.,2011. Optimal proportional reinsurance and invest-ment in a stock market with ornstein-uhlenbeck process. Insurance:Mathematics and Economics 49 (2),207-215.
[53]Lions, P. L., Sznitman, A. S.,1984. Stochastic differential equations with reflecting boundary conditions. Communications on Pure and Applied Mathematics 37 (4), 511-537.
[54]Liu, C. S., Yang, H.,2004. Optimal investment for an insurer to minimize its proba-bility of ruin. North American Actuarial Journal 8 (2),11-31.
[55]L(?)kka, A., Zervos, M.,2008. Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance:Mathematics and Economics 42 (3), 954-961.
[56]Lu, Y.,2006. On the severity of ruin in a markov-modulated risk model. Scandinavian Actuarial Journal 2006 (4),183-202.
[57]Lu, Y., Li, S.,2005. On the probability of ruin in a markov-modulated risk model. Insurance:Mathematics and Economics 37 (3),522-532.
[58]Lu, Y., Li, S.,2009. The markovian regime-switching risk model with a threshold dividend strategy. Insurance:Mathematics and Economics 44 (2),296-303.
[59]Meng, H., Siu, T.,2011. Optimal mixed impulse-equity insurance control problem with reinsurance. SIAM Journal on Control and Optimization 49 (1),254-279.
[60]Meng, H., Siu, T. K.,2011. On optimal reinsurance, dividend and reinvestment strate-gies. Economic Modelling 28 (1),211-218.
[61]Moore, K. S., Young, V. R.,2003. Pricing equity-linked pure endowments via the principle of equivalent utility. Insurance:Mathematics and Economics 33 (3),497-516.
[62]Musiela, M., Zariphopoulou, T.,2004. An example of indifference prices under expo-nential preferences. Finance and Stochastics 8 (2),229-239.
[63]Ng, A. C, Yang, H.,2006. On the joint distribution of surplus before and after ruin under a markovian regime switching model. Stochastic Processes and Their Applica-tions 116 (2),244-266.
[64]Peng, X., Chen, M., Guo, J.,2012. Optimal dividend and equity issuance problem with proportional and fixed transaction costs. Insurance:Mathematics and Economics 51 (3),576-585.
[65]Reinhard, J., Snoussi, M.,2001. On the distribution of the surplus prior to ruin in a discrete semi-markov risk model. ASTIN Bulletin 31 (2),255-274.
[66]Reinhard, J., Snoussi, M.,2002. The severity of ruin in a discrete semi-markov risk model. Stochastic Models 18 (1),85-107.
[67]Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.,2009. Stochastic processes for insurance and finance. Vol.505. Wiley, com.
[68]Schmidli, H.,2001. Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal 2001 (1),55-68.
[69]Schmidli, H.,2002. Asyrnptotics of ruin probabilities for risk processes under optimal reinsurance policies:the small claim case. Working paper 180, Laboratory of Actuarial Mathematics, University of Copenhagen.
[70]Schmidli, H.,2002. On minimizing the ruin probability by investment and reinsurance. The Annals of Applied Probability 12 (3),890-907.
[71]Shiu, E. S.,1989. The probability of eventual ruin in the compound binomial model. ASTIN Bulletin 19 (2),179-190.
[72]Sotomayor, L. R., Cadenillas, A.,2011. Classical and singular stochastic control for the optimal dividend policy when there is regime switching. Insurance:Mathematics and Economics 48 (3),344-354.
[73]Taksar, M., Markussen, C.,2003. Optimal dynamic reinsurance policies for large insurance portfolios. Finance and Stochastics 7 (1),97-121.
[74]Taksar, M., Zhou, X.,1998. Optimal risk and dividend control for a company with a debt liability. Insurance:Mathematics and Economics 22 (1),105-122.
[75]Tan, J., Yang, X.,2006. The compound binomial model with randomized decisions on paying dividends. Insurance:Mathematics and Economics 39 (1),1-18.
[76]Tremblay, L.,1992. Using the poisson inverse gaussian in bonus-malus systems. ASTIN Bulletin 22 (1),97-106.
[77]Wagner, C.,2001. A note on ruin in a two state markov model. ASTIN Bulletin 31 (2),349-358.
[78]Wagner, C.,2002. Time in the red in a two state markov model. Insurance:Mathe-matics and Economics 31 (3),365-372.
[79]Wei, J., Yang, H., Wang, R.,2010. Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. Journal of Optimization Theory and Applications 147 (2),358-377.
[80]Willmot, G. E.,1993. Ruin probabilities in the compound binomial model. Insurance: Mathematics and Economics 12 (2),133-142.
[81]Xiao, Y., Guo, J.,2007. The compound binomial risk model with time-correlated claims. Insurance:Mathematics and Economics 41 (1),124-133.
[82]Yang, H., Zhang, L.,2005. Optimal investment for insurer with jump-diffusion risk process. Insurance:Mathematics and Economics 37 (3),615-634.
[83]Young, V. R.,1999. Optimal insurance under wang's premium principle. Insurance: Mathematics and Economics 25 (2),109-122.
[84]Young, V. R.,2003. Equity-indexed life insurance:pricing and reserving using the principle of equivalent utility. North American Actuarial Journal 7 (1),68-86.
[85]Young, V. R., Zariphopoulou, T.,2002. Pricing dynamic insurance risks using the principle of equivalent utility. Scandinavian Actuarial Journal 2002 (4),246-279.
[86]Yuen, K. C., Guo, J.,2001. Ruin probabilities for time-correlated claims in the com-pound binomial model. Insurance:Mathematics and Economics 29 (1),47-57.
[87]Yuen, K. C., Guo, J.,2006. Some results on the compound markov binomial model. Scandinavian Actuarial Journal 2006 (3),129-140.
[88]Zhang, X.,2008. On the ruin problem in a markov-modulated risk model. Methodol-ogy and Computing in Applied Probability 10 (2),225-238.
[89]Zhang, X. L., Zhang, K. C., Yu, X. J.,2009. Optimal proportional reinsurance and investment with transaction costs, i:Maximizing the terminal wealth. Insurance: Mathematics and Economics 44 (3),473-478.
[90]Zhou, M., Yuen, K. C.,2012. Optimal reinsurance and dividend for a diffusion model with capital injection:Variance premium principle. Economic Modelling 29 (2),198-207.
[91]Zhou, X.,2005. On a classical risk model with a constant dividend barrier. North American Actuarial Journal 9 (4),95-108.
[92]Zhu, J., Yang, H.,2008. Ruin theory for a markov regime-switching model under a threshold dividend strategy. Insurance:Mathematics and economics 42 (1),311-318.
[93]Zhu, J., Yang, H.,2009. On differentiability of ruin functions under markov-modulated models. Stochastic Processes and their Applications 119 (5),1673-1695.