马氏调节过程在保险与金融中的应用
摘要
相对于经典的金融保险模型而言,马氏调节的金融保险模型似乎更能适应现实中的金融保险数据。在风险理论中,马氏调节的风险模型有这样一个优点:保险公司可以随外界环境(天气,经济,政府政策等)的改变而调节自身的保险政策。举个例子来说吧,在汽车保险中,天气环境的好坏是影响事故发生的重要因素。在不同的天气环境下,汽车保险中索赔的分布以及索赔在一定时间内发生的强度将会有很大的不同。因此,不同的天气环境下,保险公司的保险政策也将会有很大的不同,比如说,保费的收取将会随天气环境的变化而变化。在金融理论中,著名的Black-Scholes-Merton金融市场是基于几何布朗运动来描述标的资产(股票)的价格变化的。但是越来越多的实证研究表明,几何布朗运动并不能描述一些标的资产价格数据中的重要实证结果,比如,标的资产价格分布的重尾性质,标的资产价格的方差应该是随时间变化而变化的等等。Hardy[75]对马氏调节的金融市场模型与其他模型对现实金融数据的适合程度进行了比较并得出结论,马氏调节的金融市场模型对现实金融数据的适合程度明显优于其他一些比较常用金融市场模型。基于以上原因,马氏调节模型在金融保险理论中正变得越来越重要。
基于上面的背景以及马氏调节模型在金融保险中愈来愈重要的地位,我的博士论文致力于研究马氏调节的随机过程在保险与金融中的应用。本篇论文的结构是按如下章节安排的。
在第一章中,我们先对马氏调节模型在保险与金融历史背景及研究现状作了一个简要的回顾,然后我们详细价绍了本篇论文各章节的主要内容与所得到的主要结果。
第二章是本篇论文的理论基础。在这一章中,我们回顾了一些关于双鞅(double martingales),连续时间马氏链以及马氏调节的Levy过程的基本定义与结论。在这一章里,我们在Elliott et al.[51]的框架下引入连续时间马氏链并通过标值点过程(marked point process)的理论来刻划马氏链。基于马氏链的标值点过程刻划,我们引入了j~(th)马氏跳过程鞅的概念。j~(th)马氏跳过程鞅的引入可以说是本篇博士论文的一个重要创新。这个概念是解决第四章与第五章所考虑的马氏调节金融市场完全化问题的重要理论基础。同时,由于j~(th)马氏跳过程鞅的引入,我们可以给出关于马氏链的函数的积分表达式。此外,我们还给出了一个更加明确的关于一般马氏调节随机过程的It(?)公式。该It(?)公式在第四章刻划马氏调节布朗运动的测度变化以及随后几章中证明随机控制问题中的验证定理都起着十分重要的作用。
在余下的章节里,我的博士论文主要致力于解决马氏调节模型在金融保险应用中的五个问题。
●马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数。风险理论在Gerber and Shiu[66]于1998年引入Gerber-Shiu期望折现罚金函数后得到了长足的发展。由于一些基本的精算量,比如,破产时间,破产前余额,破产赤字都被Gerber-Shiu折现罚金函数所蕴含,因此在破产理论的历史上有大量的论文都致力于研究各种各样风险模型的Gerber-Shiu折现罚金函数的明确表达形式。Wang and Wu[163]考虑了常利率下扩散干扰的古典风险模型,Zhang et al.[178]研究了具有两步保费率的古典风险模型,Landriaultand Willmot[99],Willmot[165],Willmot and Dickson[166]则考虑了更新风险模型,而Garrido and Morales[63],Morales[122],Morales and Olivares[123]研究了Levy风险模型等等。尽管马氏调节风险模型由Janssen[88]首次引入,随后又受到了Janssen and Reinhard[89],Reinhard[140],Asmussen[5,7],B(a|¨)uerle[14],Wu[169],Wu and Wei[167],Ng and Yang[128,129],等众多学者了研究,但是Albrecher and Boxma[2]是第一个研究马氏相依风险模型的Gerber-Shiu折现罚金函数。随后Ng and Yang[129]考虑了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题,但是,他们只是给出了该模型的Gerber-Shiu期望折现罚金函数在保费率为1的情况下所满足的积分微分方程组,并没有深入讨论积分微分方程组解的情况。受Albrecher and Boxma[2]和Ng and Yang[129]这两篇文章的启发,我们考虑了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题,并且得到了此模型下Gerber-Shiu期望折现罚金函数的明确表现形式,具体结果参阅本论文的第3章。在我们之后,Lu and Tsai[110]考虑了马氏调节的扩散干扰古典风险模型的Gerber-Shiu期望折现罚金函数问题并得到了一些明确结果。在第3章中,我们充分研究了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题。我们首次给出了对于任意保费率Gerber-Shiu折现罚金函数所满足的积分微分方程组。通过Laplace变换的方法,我们得到了关于Gerber-Shiu折现罚金函数的Laplace变换的明确表达式。当索赔分布为K_n类时,通过引入Dickson and Hipp[40]文中的算子T_r并对Gerber-Shiu折现罚金函数的Laplace变换作逆变换,我们得到了Gerber-Shiu折现罚金函数的明确表达式。最后,我们给出了两个具体的数值例子来更加直观的说明我们的结论。
●利用双鞅(double martingales)理论完全化马氏调节几何布朗运动金融市场(Markov-modulated geometric Brownian motion market)的方法。在金融理论的发展历史中,金融市场的完全性是一个十分重要的概念。与不完全金融市场理论相比,完全金融市场理论更加成熟。期权定价,最大化期望效用的投资等问题在完全化的金融市场里都已得到完全解决,而相应问题在非完全市场中仍未完全解决。因此,如果我们能找到将不完全金融市场完全化的方法,我们便可以利用完全金融市场中的结论来解决不完全金融市场中最初所要解决的问题。Karatzas et al.[95]通过增加假想资产来扩大不完全金融市场达到完全化市场的目的,并利用完全金融市场中的最大化期望效用问题的结论解决了最初不完全金融市场中的最大化期望效用问题。马氏调节几何布朗运动金融市场是不完全的,这一点可以从Elliott and Swishchuk[48]论文中得到。因此我们希望可以找到一个方法来完全化马氏调节几何布朗运动金融市场并考虑完全化后的马氏调节几何布朗运动金融市场中的最优资产组合问题。之所以考虑完全化马氏调节几何布朗运动金融市场是基于下面四篇文章的启发:Corcuera et al.[34],Corcuera et al.[35],Niu[130]和Guo[71]。通过添加幂跳资产(power-jump asset),Corcuera et al.[34]实现了不完全几何Levy金融市场的完全化。Corcuera et al.[35]则考虑了完全化后的几何Levy金融市场中的最优资产组合以达到最大期望效用问题。Niu[130]则利用Corcuera et al.[34]中的方法来完全化简单Levy金融市场并考虑完全化后的金融市场中如何进行投资分配以达到最大期望效用问题。Guo[71]虽然给出了一种方法来完全化马氏调节几何布朗运动金融市场,但是这种方法很难用于考虑完全化后的马氏调节几何布朗运动金融市场中的最大化期望效用问题。因此我们需要找到其他的方法来完全化马氏调节几何布朗运动金融市场并考虑完全化后市场中如何进行投资分配以达到最大期望效用。
在第4章与第5章中,我们以双鞅理论与标值点过程理论作为理论基础解决了马氏调节几何布朗运动金融市场完全化问题。利用动态规划理论方法,我们解决了完全化马氏调节几何布朗运动金融市场中的最大化期望效用的投资组合问题。
在第4章中,我们首次明确给出马氏调节布朗运动在等价鞅测度变换下二者参数之间的关系式。利用Elliott[44]文中的双鞅表示理论,我们首次通过添加j~(th)马氏跳资产(j~(th) Markov jump assets)的方法实现了马氏调节几何布朗运动金融市场的完全化。我们严格证明了添加j~(th)马氏跳资产后的马氏调节几何布朗运动金融市场是完全的。对于完全化后的马氏调节几何布朗运动金融市场中非负权益的对冲策略给出了明确表现形式。我们也证明了完全化后的马氏调节几何布朗运动金融市场是无套利市场,并且该完全市场中唯一的等价鞅测度也明确给出。
在第5章中,我们对第4中的完全化方法作了一个小的变动。在第4章中的完全化方法中,一个潜在的问题就是:j~(th)马氏跳资产以一个正的概率取值为负。为了将此问题排除在外,我们引入j~(th)马氏几何跳资产(j~(th) Markoviangeometric jump assets)。我们证明了添加了j~(th)马氏几何跳资产后的金融市场是完全的并且是无套利的,其相应的唯一等价鞅测度也给出了明确的表现形式。在完全化的马氏调节几何布朗运动金融市场中,我们很好的解决了最大化对数与幂函数效用的投资组合问题。在考虑对数效用问题中,我们通过直接微分方法得到了最优策略与最优值函数的显式表达。而在考虑幂函数效用时,我们则通过动态规划的方法得到了相应的最优策略与最优值函数的显式表达。此外,在两种效用函数的投资组合问题中,我们还给出了最优值函数的Feynman-Kac表现形式。
●卖空限制与具有多个风险资产的扩展马氏调节金融市场(Extended Markovianregime-switching market)下最大化期望效用的再保险与投资问题。期望效用作为金融保险中的一个重要目标函数,得到了人们越来越多的关注,参考论文Bai and Guo[10,11],Brendle[23],Browne[25],Cvitanic andKaratzas[37],Karatzas et al.[95],Lakner[97],Pham and Quenez[132],Yangand Zhang[172]等等。Browne[25]首次考虑了扩散模型下最大化指数效用的投资策略与最小化破产概率的投资策略之间的关系。Bai and Guo[10]则将Browne[25]的结果推广到具有卖空限制和具有多个风险资产的金融市场情形下,而在Bai and Guo[11]文中则考虑了局部信息下最大化期望效用问题。Yang and Zhang[172]则研究了跳扩散风险模型下最优投资组合问题并得到了最优策略与最优值函数的显式表达。
相比以上文章,我们则考虑马氏调节模型框架下的最大化期望效用的再保险与投资问题。保险公司的盈余过程与金融市场都是受马氏环境调控的。在第6与第7章中所考虑的金融市场是扩展马氏调节金融市场。扩展马氏调节金融市场与第4章与第5章所考虑的马氏调节几何布朗运动金融市场有如下两个主要区别:首先扩展马氏调节金融市场由一个无风险资产与d个风险资产组成的多资产金融市场;其次无风险债券利率,风险资产的回报率与波动率不仅仅由马氏链控制,而且还随时间t的变动而变动。关于扩展马氏调节金融市场的描述可参阅本论文的6.2节内容。
在第6章中,我们单纯考虑了扩展马氏调节金融市场中的最大期望效用的投资组合问题。利用直接微分法与动态规划方法,我们解决了扩展马氏调节金融市场最大化对数与幂函数效用问题。在有卖空限制与没有卖空限制两种情况下,我们都得到了最优投资策略的显式表达与最优值函数的Feynman-Kac表现形式。
相对于第6章,我们在第7章中增加了保险模型并考虑了保险公司最优比例再保险与扩展马氏调节金融市场中的投资问题。与传统的最优比例再保险与投资问题(参考文章Browne[25])相比,我们所考虑的问题主要有下面几个改进:首先,我们用马氏调节的扩散过程来描述保险公司的盈余过程;其次,所投资的金融市场是扩展马氏调节金融市场,该市场中具有多个风险资产;再者,我们所考虑的投资问题是不准卖空风险资产限制下的投资。我们以最大化指数效用为目标,利用动态规划原则中的HJB(Hamilton-Jacobi-Bellman)方程得到了在不准卖空风险资产限制下最优比例再保险与投资策略的显式表达。我们首先给出最优值函数所满足的Hamilton-Jacobi-Bellman(HJB)方程,然后通过一般的分离变量法构造出HJB方程的显式解。最后,通过验证定理,我们得到了我们所构造的HJB方程的解的确就是我们所要寻找的最优值函数。在这一章中最优再保险与投资策略以及最优值函数都给出了明确表达式,除此之外,我们还得到了Bai and Cuo[10]文中第一部分中的内容是我们的一个特殊情况。我们利用我们的结果验证了Bai and Guo[10]文中的结论。
●马氏调节跳扩散市场(Markov-switching jump-diffusion market)下的均值-方差问题。与期望效用最大化问题相比,均值-方差问题可以让投资者在自已能够承受的风险范围内最大化自已的收益,而期望效用问题只是单独考虑效用最大化而没有将效用的风险考虑在内。均值-方差问题最初由Markowitz[113,114]提出并研究了单期市场的均值-方差问题。经过几十年的发展,现在均值-方差理论已成为现代金融理论的基础并启发了大量的扩展与应用。连续时间几何布朗运动市场中的均值-方差问题可以说是Markowitz均值方差理论的一个重要推广,在过去的十几年里得到了大量学者的关注,参考文章Bielecki et al.[20],Li et al.[102],Lim[103],Lim and Zhou[104],Xiong andZhou[170],Zhou and Li[181],等等。然而,马氏调节金融市场中的均值-方差问题却很少有人研究,这主要是由于马氏调节金融市场中增加了不确定因素-马氏链来描述外界环境的变化,从而导致了解决此类问题的难度加大。据我们所知,在均值-方差理论的研究历史中只有三篇文章考虑了马氏调节金融市场中的此类问题,它们分别是Zhou and Yin[182],Yin and Zhou[174]and Chen et al.[31]。但是,在这三篇文章中所考虑的马氏调节金融市场是无跳连续的,而越来越多的实际数据证明,带跳的金融市场能更好的模拟现实中的金融市场。因此在8章中,我们致力于研究马氏调节跳扩散市场中均值-方差问题。
在第8章中,我们首先给出了马氏调节跳扩散市场的概念,然后利用Zhouand Li[181]文中的方法将均值-方差问题转换成为随机线性二次规划问题。我们讨论了马氏调节跳扩散市场下均值-方差问题的可行性(feasibility)并给出了均值-方差问题可行性的几个等价条件。通过动态规划中的HJB方程方法,我们解决了转换后的随机线性二次规划问题。利用随机线性二次规划问题与最初的均值-方差问题之间的联系,我们得到了均值-方差问题中的有效组合与有效前沿的明确表达式。我们还研究了最小方差以及与其相应的投资组合并得到了明确的结果。此外,我们还证明了共同基金定理在马氏调节跳扩散市场下仍然成立。
●隐马尔科夫模型(Hidden Markov model)中的最优再保险与投资问题。最近,越来越多的学者开始考虑金融保险中的隐马尔科夫模型中投资与再保险问题,例如B(a|¨)uerle and Rieder[17],Elliott and van der Hoek[49],Elliott et al.[51],Lakner[97],Nagai and Runggaldier[126],Pham andQuenez[132],Rieder and B(a|¨)uerle[141],Sass[146],Sass and Haussmann[147]等。Lakner[97]利用鞅方法研究了当金融市场中股票回报率无法观测情况下的最优投资与消费问题并且对对数效用与幂效用得到了显式表达。Lakner[98]利用同样的方法解决了股票回报率为线性扩散过程时相应的问题。B(a|¨)uerle andRieder[17],Sass and Haussmann[147]考虑了当金融市场中股票的回报率受一投资者无法观测的马氏链调节情况下最大化期望效用的投资组合问题。B(a|¨)uerle and Rieder[17]则考虑了带跳金融市场中无法观测跳的强度下的最大化期望效用的投资组合问题。Nagai and Runggaldier[126]则通过偏微分方程理论研究了隐马尔科夫模型中最大化期望效用的投资组合问题。
纵观以上文章,我们发现它们都只是考虑隐马尔科夫模型中的最优投资问题,并没有涉及再保险问题。因此在第9章中,我们将考虑保险公司在局部市场信息下,如何采取再保险与投资策略来最大化终值指数效用。此处局部信息是指,对于股票市场,保险公局只能知道股票价格的信息,而相应的股票的回报率是受一无法观测的马氏链控制,即此处的市场为隐马尔科夫金融市场。在此章中,保险公司的盈余过程由带漂移的布朗运动来描述,而所考虑的隐马尔科夫金融市场是一个具有多个风险资产的市场。通过建立分离原理我们将原有的局部信息问题转化成完全信息问题。利用动态规划原则以及Girsanov变换方法,我们得到最优值函数是一线性抛物偏微分方程的唯一粘性解。
Compared with the classical risk model or finance model, the Markov-modulated model or Markovian regime-switching model seems to provide a better fit to the reality data of insurance and finance. In risk theory, the Markov-modulated risk model can capture the feature that insurance policies may 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 process in different weather conditions will be very different. Therefore the insurance policies of insurance companies, such as premium rate, will be different in different weather conditions. In finance theory, the celebrated Black-Scholes-Merton financial market is based on a geometric Brownian motion to capture the price dynamics of the underlying security. However, numerous empirical studies reveal that this assumption for assets price dynamics cannot provide realistic description for some important empirical behavior of financial returns such as the heavy-tailedness of the return's distribution and the time-varying volatility of return. Hardy [75] shows that the Markovian regime-switching model provides a significantly better fit to the data than do other popular models. Therefore, the class of Markov-modulated or Markovian regime-switching model becomes more and more important in insurance and finance.
On the basis of these background, my doctoral dissertation is mainly devoted to considering the applications of Markov-modulated or Markovian regime-switching model in insurance and finance. The dissertation is organized as follows.
A brief overview of the history of Markov-modulated or Markovian regime-switching model in risk theory and finance theory, as well as the main contents of this dissertation, are given in the first chapter.
The second chapter is the theoretical foundation of this dissertation. In this chapter, we review some basic definitions and results on double martingales, continuous time Markov chain and Markov-modulated Lévy processes. The main contribution of this chapter is that we introduce the Markov chain following the framework of Elliott et al. [51] and characterize the Markov chain by associating it with a marked point process. Based on the marked point process characterization of Markov chain, we introduce the j~(th) Markov jump martingale which plays a major rule in completing the Markovian regime-switching market in Chapter 4. Besides, we also derive some new results, such as the integral representation of the function of Markov chain and the It(o|^) differential rule for the generalized Markov-modulated stochastic process, which are very useful in characterizing the measure change of Markov-modulated Brownian motion in Chapter 4 and proving the verification theorem for the optimization problems in the sequent chapters.
The remaining chapters are devoted to solving the following five problems of the Markov-modulated or Markovian regime-switching model in insurance and finance.
Gerber-Shiu discounted penalty function of Markov-modulated compound Poisson risk model. Risk theory received a substantial boost with the article of Gerber and Shiu [66] in 1998, in which the Gerber-Shiu discounted penalty function was introduced. Since the basic actuarial variables such as the time of ruin, the surplus immediately prior to ruin, the deficit at ruin, are embedded in the Gerber-Shiu discounted penalty function, a large number of papers in the ruin literature are devoted to obtaining closed forms of Gerber-Shiu discounted penalty functions in a variety of risk models, see Wang and Wu [163] for the perturbed compound Poisson risk process with constant interest, Zhang et al. [178] for classical risk model with a two-step premium rate, Landriault and Willmot [99], Willmot [165], Willmot and Dickson [166] for the renewal risk model, Garrido and Morales [63], Morales [122], Morales and Olivares [123] for the Lévy risk model. Although the Markov-modulated risk model was first introduced by Janssen [88] and later treated by Janssen and Reinhard [89], Reinhard [140], Asmussen [5, 7], B(a|¨)uerle [14], Wu [169] ,Wu and Wei [167], Ng and Yang [128, 129] and so on, Albrecher and Boxma [2] were the first to consider the Gerber-Shiu discounted penalty function for the Markov-dependent risk model. Later Ng and Yang [129] also considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. However, they only gave a system of integro-differential equations for Gerber-Shiu functions and did not discuss the solution of this system of integro-differential equations. Inspired by the work of Albrecher and Boxma [2] and Ng and Yang [129], we study the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model and obtain closed forms of Gerber-Shiu discounted penalty functions for this model, see Chapter 3 of my doctoral dissertation. Later Lu and Tsai [110] considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model perturbed by diffusion.
Chapter 3 is devoted to obtaining closed forms of the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. We first present a system of integro-differential equations of the Gerber-Shiu discounted penalty functions for arbitrary premium rate. By using Laplace transform to solve the integro-differential equations, we obtain explicit expressions of the Laplace transforms of Gerber-Shiu discounted penalty functions. Closed forms of the discounted joint density function of surplus prior to and after ruin for the initial surplus 0 are obtained. Besides, we derive explicit formulas of Gerber-Shiu discounted penalty functions for the K_n-family claim size distributions in the two-state case through introducing the operator T_r. Finally, numerical examples are presented to illustrate our results.
Completing the Markovian regime-switching market via double martingales. Completeness of the financial market is an important concept in the finance literature and the theory of the complete market is more mature than that of the incomplete market. Therefore, if we can find a method to complete the incomplete market, we can use the results of the complete market to solve the problem in the original incomplete market. Karatzas et al. [95] introduced a way to complete the incomplete market by adding the fictitious stocks for the original incomplete market. By using the results of utility maximization problem in the complete market and the relationship between the complete market and the original incomplete market, Karatzas et al. [95] solved the utility maximization problem in the original incomplete market. The Markovian regime-switching market consists of one bond and one stock, see Chapter 4 for the description of this market. It is well-known that the Markovian regime-switching market is incomplete (see Elliott and Swishchuk [48]). So we want to find a method to complete the Markovian regime-switching market and study the portfolio optimization in the completed Markovian regime-switching market. The idea of completing the Markovian regime-switching is inspired by Corcuera et al. [34], Corcuera et al. [35], Niu [130] and Guo [71]. By adding the power-jump assets (related to the suitably compensated power-jump processes of the underlying Lévy process), Corcuera et al. [34] completed the geometric Lévy market, while, Corcuera et al. [35] considered the optimal investment problems in the completed geometric Lévy market with the new power-jump assets. Niu [130] studied the optimal investment problems in a simple Lévy market which was completed by adding the power-jump assets introduced by Corcuera et al. [34]. Guo [71] introduced the change-of-state contract to complete the Markovian regime-switching market. However, this method is not good enough for the solution of optimal investment problems. Therefore we need to find some other method to complete the market and consider the optimal investment problems.
In Chapter 4 and Chapter 5, we find two new methods to complete the Markovian regime-switching market via the theory of double martingales and marked point process. The problem of portfolio optimization for maximizing expected utility in the completed Markovian regime-switching market is well solved.
In Chapter 4, we first give a characterization of all structure-preserving equivalent martingale measure, under which the Markov-modulated Brownian motion remains a Markov-modulated Brownian motion but with changed parameters. Then by using the martingale representation technique of double martin- gales introduced by Elliott [44], we complete the Markovian regime-switching market by adding the j~(th) Markovian jump assets. We prove that the enlarged Markovian regime-switching market(i.e. the Markovian regime-switching market completed by adding the j~(th) Markovian jump assets.) is complete and the explicit hedging policy for a non-negative square-integrable contingent in the enlarged Markovian regime-switching market is also obtained. We also prove that the enlarged Markovian regime-switching market is arbitrage-free and the unique equivalent martingale measure is determined by the explicit expression of likelihood ratio process.
In Chapter 5, we make a minor modification to the method of completing the Markovian regime-switching market in Chapter 4. One potential problem of the j~(th) Markovian jump assets introduced in Chapter 4 is that it can take negative values with positive probability. To exclude this potential problem, we consider a set of j~(th) Markovian geometric jump assets, which always take non-negative values. We prove that the Markovian regime-switching market enlarged by a set of j~(th) Markovian geometric jump assets is complete and arbitrage-free. The portfolio selection problem in the enlarged market in the case of a power utility and a logarithmic utility is well solved. In the case of logarithmic utility we adopt the direct differentiation approach to derive the closed-form solution of the optimal portfolio strategy, while for the power utility, we adopt the dynamic programming approach to derive the optimal portfolio strategies. Closed-form solutions for the optimal portfolio strategies and the value function are obtained in both cases. Besides, we also give a Feynman-Kac representation of the value function in both utility cases.
Maximizing the expected utility with proportional reinsurance and investment in the extended Markovian regime-switching market with multiple risky assets under no-shorting constraint. Expected utility, as an important objective function in the financial and actuarial literature, has attracted a great deal of interest, see Bai and Guo [10, 11], Brendle [23], Browne [25], Cvitanic and Karatzas [37], Karatzas et al. [95], Lakner [97], Pham and Quenez [132], Yang and Zhang [172] and so on. Browne [25] considered the optimal investment policies for maximizing exponential utility and minimizing the probability of ruin. Bai and Guo [10] extended the result of Browne [25] to the case of the financial market with multiple risky asset and no-shorting constraint. Bai and Guo [11] also considered the utility maximization in the case of partial information. Yang and Zhang [172] studied the optimal investment policies of an insurer with jump-diffusion risk process. The explicit expressions for the optimal strategy and the value function are given.
In contrast to above papers, we consider the problem of maximizing the expected utility in the Markov-modulated case, i.e., the surplus of the insurance company and the prices of the financial market are modulated by a continuous time Markov chain. The financial market considered in Chapter 6 and Chapter 7 is the extended Markovian regime-switching market. There are two main differences between the extended Markovian regime-switching market and the Markovian regime-switching market considered in Chapter 4 and Chapter 5: firstly, the extended Markovian regime-switching market consists of one bond and d stocks while the Markovian regime-switching market considered in Chapter 4 and Chapter 5 only consists of one bond and one stock; secondly, in the extended Markovian regime-switching market the bank interest rates, stocks' appreciation rates, and volatility rates not only are modulated by a continuous Markov chain, but also depend on time t.
In Chapter 6, we adopt the direct differentiation approach and dynamic programming approach to solve the portfolio selection problems in the extended Markovian regime-switching market for the logarithmic and power utility respectively. Closed-form solutions for the optimal portfolio strategies and the Feynman-Kac representation of value functions in both logarithmic and power utility with no shorting constraint or not are obtained.
In Chapter 7, we focus on the optimal proportional reinsurance and investment in the extended Markovian regime-switching market. The main differences from the classical optimal proportional reinsurance and investment problems (see Browne [25]) are: firstly, the surplus of the insurance company is modeled by a Markov-modulated Brownian motion; secondly, the financial market is the extended Markovian regime-switching market; thirdly, no shorting constraint is considered in our problem. We adopt the dynamic programming approach to obtain the optimal proportional reinsurance and investment strategy for maximizing exponential utility function from terminal wealth under the no-shorting constraint. We first present the Hamilton-Jacobi-Bellman (HJB) equation satisfied by the optimal value function and then construct the solution of the HJB equation by the usual separable variable approach. At last, we verify that the solution of the HJB equation is indeed the optimal value function. Closed-form of the optimal proportional reinsurance and investment strategies and the Feynman-Kac representation of the optimal value function are obtained. Besides, we find that the first part of Bai and Guo [10] is a special case of our results. We verify the result of Bai and Guo [10] through our model.
Mean-Variance problem in the Markov-switching jump-diffusion market. Compared with the utility maximization problems, mean-variance problem enables an investor to seek the highest return after specifying his/her acceptable risk level that is quantified by the variance of the return. The mean-variance problem was originally proposed by Markowitz [113, 114] for portfolio construction in a single period. Now the mean-variance approach has become the foundation of modern finance theory and has inspired numerous extensions and applications. The continuous time mean-variance problem in the geometric Brownian motion market has been considered by many authors, see Bielecki et al. [20], Li et al. [102], Lim [103], Lim and Zhou [104], Xiong and Zhou [170], Zhou and Li [181], and so on. However, the papers on the mean-variance problem in the Markovian regime-switching market are quite few. So far as we know, there are only three papers on the mean-variance problem in the Markovian regime-switching market, see Zhou and Yin [182], Yin and Zhou [174] and Chen et al. [31]. The above three papers only consider mean-variance problem in the case of the Markovian regime-switching market without jump. So in Chapter 8, we consider mean-variance problem in the Markov-switching jump-diffusion market.
In Chapter 8, we first present the description of the Markov-switching jump- diffusion market and reduce the mean-variance problem to a stochastic linear-quadratic (LQ) problem by using the method of Zhou and Li [181]. The feasibility of the mean-variance problem is discussed and some necessary and sufficient conditions for the feasibility in Markov-switching market are given. Through dynamic programming technique, we solve the stochastic LQ problem and obtain the closed form of the value function. Prom the relationship between the stochastic LQ problem and the original mean-variance problem and the well-known Lagrange duality theorem, we obtain efficient portfolio and efficient frontier in a closed form. We also obtain the minimum variance, i.e. the minimum possible terminal variance, along with the portfolio that attains the minimum variance. Besides, we also prove that the mutual fund theorem in the Markov-switching market still holds.
Optimal reinsurance and investment in a hidden Markov model. Recently, more and more papers consider the hidden Markov model in insurance and finance, see Bauerle and Rieder [17], Elliott and van der Hoek [49], Elliott et al. [51], Lakner [97], Nagai and Runggaldier [126], Pham and Quenez [132], Rieder and Bauerle [141], Sass [146], Sass and Haussmann [147], and so on. Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Bauerle and Rieder [17], Sass and Haussmann [147] considered the problem of maximizing the expected utility of the terminal wealth in the case that the drift rate of the stock is Markov-modulated and can not be observed by the investor. Bauerle and Rieder [17] studied the hidden Markov jump intensity model. Using a classical result from filter theory, the hidden problem was reduced to the complete observation problem. Nagai and Runggaldier [126] considered the utility maximization for market models with hidden markov factors through PDE approach.
Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Pham and Quenez [132] addressed the maximization problem of expected utility from terminal wealth in a financial market where price process of risky assets follows a stochastic volatility model and only the vector of stock prices are observed by the investors.
Based on these papers, in Chapter 9, we consider an optimal reinsurance and investment problems in a multiple risky assets market with appreciation rate driven by a hidden Markov chain. The surplus of the insurance company is modeled by a Brownian motion with drift and the objective function is the expected exponential utility. By using the filter theory, we establish the separation principle and reduce the problem to the completely observed case. Through the dynamic programming approach and the Girsanov change of measure, we characterize the value function as the unique viscosity solution of a linear parabolic partial differential equation and obtain the Feynman-Kac representation of the value function.
基于上面的背景以及马氏调节模型在金融保险中愈来愈重要的地位,我的博士论文致力于研究马氏调节的随机过程在保险与金融中的应用。本篇论文的结构是按如下章节安排的。
在第一章中,我们先对马氏调节模型在保险与金融历史背景及研究现状作了一个简要的回顾,然后我们详细价绍了本篇论文各章节的主要内容与所得到的主要结果。
第二章是本篇论文的理论基础。在这一章中,我们回顾了一些关于双鞅(double martingales),连续时间马氏链以及马氏调节的Levy过程的基本定义与结论。在这一章里,我们在Elliott et al.[51]的框架下引入连续时间马氏链并通过标值点过程(marked point process)的理论来刻划马氏链。基于马氏链的标值点过程刻划,我们引入了j~(th)马氏跳过程鞅的概念。j~(th)马氏跳过程鞅的引入可以说是本篇博士论文的一个重要创新。这个概念是解决第四章与第五章所考虑的马氏调节金融市场完全化问题的重要理论基础。同时,由于j~(th)马氏跳过程鞅的引入,我们可以给出关于马氏链的函数的积分表达式。此外,我们还给出了一个更加明确的关于一般马氏调节随机过程的It(?)公式。该It(?)公式在第四章刻划马氏调节布朗运动的测度变化以及随后几章中证明随机控制问题中的验证定理都起着十分重要的作用。
在余下的章节里,我的博士论文主要致力于解决马氏调节模型在金融保险应用中的五个问题。
●马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数。风险理论在Gerber and Shiu[66]于1998年引入Gerber-Shiu期望折现罚金函数后得到了长足的发展。由于一些基本的精算量,比如,破产时间,破产前余额,破产赤字都被Gerber-Shiu折现罚金函数所蕴含,因此在破产理论的历史上有大量的论文都致力于研究各种各样风险模型的Gerber-Shiu折现罚金函数的明确表达形式。Wang and Wu[163]考虑了常利率下扩散干扰的古典风险模型,Zhang et al.[178]研究了具有两步保费率的古典风险模型,Landriaultand Willmot[99],Willmot[165],Willmot and Dickson[166]则考虑了更新风险模型,而Garrido and Morales[63],Morales[122],Morales and Olivares[123]研究了Levy风险模型等等。尽管马氏调节风险模型由Janssen[88]首次引入,随后又受到了Janssen and Reinhard[89],Reinhard[140],Asmussen[5,7],B(a|¨)uerle[14],Wu[169],Wu and Wei[167],Ng and Yang[128,129],等众多学者了研究,但是Albrecher and Boxma[2]是第一个研究马氏相依风险模型的Gerber-Shiu折现罚金函数。随后Ng and Yang[129]考虑了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题,但是,他们只是给出了该模型的Gerber-Shiu期望折现罚金函数在保费率为1的情况下所满足的积分微分方程组,并没有深入讨论积分微分方程组解的情况。受Albrecher and Boxma[2]和Ng and Yang[129]这两篇文章的启发,我们考虑了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题,并且得到了此模型下Gerber-Shiu期望折现罚金函数的明确表现形式,具体结果参阅本论文的第3章。在我们之后,Lu and Tsai[110]考虑了马氏调节的扩散干扰古典风险模型的Gerber-Shiu期望折现罚金函数问题并得到了一些明确结果。在第3章中,我们充分研究了马氏调节复合Poisson风险模型的Gerber-Shiu期望折现罚金函数问题。我们首次给出了对于任意保费率Gerber-Shiu折现罚金函数所满足的积分微分方程组。通过Laplace变换的方法,我们得到了关于Gerber-Shiu折现罚金函数的Laplace变换的明确表达式。当索赔分布为K_n类时,通过引入Dickson and Hipp[40]文中的算子T_r并对Gerber-Shiu折现罚金函数的Laplace变换作逆变换,我们得到了Gerber-Shiu折现罚金函数的明确表达式。最后,我们给出了两个具体的数值例子来更加直观的说明我们的结论。
●利用双鞅(double martingales)理论完全化马氏调节几何布朗运动金融市场(Markov-modulated geometric Brownian motion market)的方法。在金融理论的发展历史中,金融市场的完全性是一个十分重要的概念。与不完全金融市场理论相比,完全金融市场理论更加成熟。期权定价,最大化期望效用的投资等问题在完全化的金融市场里都已得到完全解决,而相应问题在非完全市场中仍未完全解决。因此,如果我们能找到将不完全金融市场完全化的方法,我们便可以利用完全金融市场中的结论来解决不完全金融市场中最初所要解决的问题。Karatzas et al.[95]通过增加假想资产来扩大不完全金融市场达到完全化市场的目的,并利用完全金融市场中的最大化期望效用问题的结论解决了最初不完全金融市场中的最大化期望效用问题。马氏调节几何布朗运动金融市场是不完全的,这一点可以从Elliott and Swishchuk[48]论文中得到。因此我们希望可以找到一个方法来完全化马氏调节几何布朗运动金融市场并考虑完全化后的马氏调节几何布朗运动金融市场中的最优资产组合问题。之所以考虑完全化马氏调节几何布朗运动金融市场是基于下面四篇文章的启发:Corcuera et al.[34],Corcuera et al.[35],Niu[130]和Guo[71]。通过添加幂跳资产(power-jump asset),Corcuera et al.[34]实现了不完全几何Levy金融市场的完全化。Corcuera et al.[35]则考虑了完全化后的几何Levy金融市场中的最优资产组合以达到最大期望效用问题。Niu[130]则利用Corcuera et al.[34]中的方法来完全化简单Levy金融市场并考虑完全化后的金融市场中如何进行投资分配以达到最大期望效用问题。Guo[71]虽然给出了一种方法来完全化马氏调节几何布朗运动金融市场,但是这种方法很难用于考虑完全化后的马氏调节几何布朗运动金融市场中的最大化期望效用问题。因此我们需要找到其他的方法来完全化马氏调节几何布朗运动金融市场并考虑完全化后市场中如何进行投资分配以达到最大期望效用。
在第4章与第5章中,我们以双鞅理论与标值点过程理论作为理论基础解决了马氏调节几何布朗运动金融市场完全化问题。利用动态规划理论方法,我们解决了完全化马氏调节几何布朗运动金融市场中的最大化期望效用的投资组合问题。
在第4章中,我们首次明确给出马氏调节布朗运动在等价鞅测度变换下二者参数之间的关系式。利用Elliott[44]文中的双鞅表示理论,我们首次通过添加j~(th)马氏跳资产(j~(th) Markov jump assets)的方法实现了马氏调节几何布朗运动金融市场的完全化。我们严格证明了添加j~(th)马氏跳资产后的马氏调节几何布朗运动金融市场是完全的。对于完全化后的马氏调节几何布朗运动金融市场中非负权益的对冲策略给出了明确表现形式。我们也证明了完全化后的马氏调节几何布朗运动金融市场是无套利市场,并且该完全市场中唯一的等价鞅测度也明确给出。
在第5章中,我们对第4中的完全化方法作了一个小的变动。在第4章中的完全化方法中,一个潜在的问题就是:j~(th)马氏跳资产以一个正的概率取值为负。为了将此问题排除在外,我们引入j~(th)马氏几何跳资产(j~(th) Markoviangeometric jump assets)。我们证明了添加了j~(th)马氏几何跳资产后的金融市场是完全的并且是无套利的,其相应的唯一等价鞅测度也给出了明确的表现形式。在完全化的马氏调节几何布朗运动金融市场中,我们很好的解决了最大化对数与幂函数效用的投资组合问题。在考虑对数效用问题中,我们通过直接微分方法得到了最优策略与最优值函数的显式表达。而在考虑幂函数效用时,我们则通过动态规划的方法得到了相应的最优策略与最优值函数的显式表达。此外,在两种效用函数的投资组合问题中,我们还给出了最优值函数的Feynman-Kac表现形式。
●卖空限制与具有多个风险资产的扩展马氏调节金融市场(Extended Markovianregime-switching market)下最大化期望效用的再保险与投资问题。期望效用作为金融保险中的一个重要目标函数,得到了人们越来越多的关注,参考论文Bai and Guo[10,11],Brendle[23],Browne[25],Cvitanic andKaratzas[37],Karatzas et al.[95],Lakner[97],Pham and Quenez[132],Yangand Zhang[172]等等。Browne[25]首次考虑了扩散模型下最大化指数效用的投资策略与最小化破产概率的投资策略之间的关系。Bai and Guo[10]则将Browne[25]的结果推广到具有卖空限制和具有多个风险资产的金融市场情形下,而在Bai and Guo[11]文中则考虑了局部信息下最大化期望效用问题。Yang and Zhang[172]则研究了跳扩散风险模型下最优投资组合问题并得到了最优策略与最优值函数的显式表达。
相比以上文章,我们则考虑马氏调节模型框架下的最大化期望效用的再保险与投资问题。保险公司的盈余过程与金融市场都是受马氏环境调控的。在第6与第7章中所考虑的金融市场是扩展马氏调节金融市场。扩展马氏调节金融市场与第4章与第5章所考虑的马氏调节几何布朗运动金融市场有如下两个主要区别:首先扩展马氏调节金融市场由一个无风险资产与d个风险资产组成的多资产金融市场;其次无风险债券利率,风险资产的回报率与波动率不仅仅由马氏链控制,而且还随时间t的变动而变动。关于扩展马氏调节金融市场的描述可参阅本论文的6.2节内容。
在第6章中,我们单纯考虑了扩展马氏调节金融市场中的最大期望效用的投资组合问题。利用直接微分法与动态规划方法,我们解决了扩展马氏调节金融市场最大化对数与幂函数效用问题。在有卖空限制与没有卖空限制两种情况下,我们都得到了最优投资策略的显式表达与最优值函数的Feynman-Kac表现形式。
相对于第6章,我们在第7章中增加了保险模型并考虑了保险公司最优比例再保险与扩展马氏调节金融市场中的投资问题。与传统的最优比例再保险与投资问题(参考文章Browne[25])相比,我们所考虑的问题主要有下面几个改进:首先,我们用马氏调节的扩散过程来描述保险公司的盈余过程;其次,所投资的金融市场是扩展马氏调节金融市场,该市场中具有多个风险资产;再者,我们所考虑的投资问题是不准卖空风险资产限制下的投资。我们以最大化指数效用为目标,利用动态规划原则中的HJB(Hamilton-Jacobi-Bellman)方程得到了在不准卖空风险资产限制下最优比例再保险与投资策略的显式表达。我们首先给出最优值函数所满足的Hamilton-Jacobi-Bellman(HJB)方程,然后通过一般的分离变量法构造出HJB方程的显式解。最后,通过验证定理,我们得到了我们所构造的HJB方程的解的确就是我们所要寻找的最优值函数。在这一章中最优再保险与投资策略以及最优值函数都给出了明确表达式,除此之外,我们还得到了Bai and Cuo[10]文中第一部分中的内容是我们的一个特殊情况。我们利用我们的结果验证了Bai and Guo[10]文中的结论。
●马氏调节跳扩散市场(Markov-switching jump-diffusion market)下的均值-方差问题。与期望效用最大化问题相比,均值-方差问题可以让投资者在自已能够承受的风险范围内最大化自已的收益,而期望效用问题只是单独考虑效用最大化而没有将效用的风险考虑在内。均值-方差问题最初由Markowitz[113,114]提出并研究了单期市场的均值-方差问题。经过几十年的发展,现在均值-方差理论已成为现代金融理论的基础并启发了大量的扩展与应用。连续时间几何布朗运动市场中的均值-方差问题可以说是Markowitz均值方差理论的一个重要推广,在过去的十几年里得到了大量学者的关注,参考文章Bielecki et al.[20],Li et al.[102],Lim[103],Lim and Zhou[104],Xiong andZhou[170],Zhou and Li[181],等等。然而,马氏调节金融市场中的均值-方差问题却很少有人研究,这主要是由于马氏调节金融市场中增加了不确定因素-马氏链来描述外界环境的变化,从而导致了解决此类问题的难度加大。据我们所知,在均值-方差理论的研究历史中只有三篇文章考虑了马氏调节金融市场中的此类问题,它们分别是Zhou and Yin[182],Yin and Zhou[174]and Chen et al.[31]。但是,在这三篇文章中所考虑的马氏调节金融市场是无跳连续的,而越来越多的实际数据证明,带跳的金融市场能更好的模拟现实中的金融市场。因此在8章中,我们致力于研究马氏调节跳扩散市场中均值-方差问题。
在第8章中,我们首先给出了马氏调节跳扩散市场的概念,然后利用Zhouand Li[181]文中的方法将均值-方差问题转换成为随机线性二次规划问题。我们讨论了马氏调节跳扩散市场下均值-方差问题的可行性(feasibility)并给出了均值-方差问题可行性的几个等价条件。通过动态规划中的HJB方程方法,我们解决了转换后的随机线性二次规划问题。利用随机线性二次规划问题与最初的均值-方差问题之间的联系,我们得到了均值-方差问题中的有效组合与有效前沿的明确表达式。我们还研究了最小方差以及与其相应的投资组合并得到了明确的结果。此外,我们还证明了共同基金定理在马氏调节跳扩散市场下仍然成立。
●隐马尔科夫模型(Hidden Markov model)中的最优再保险与投资问题。最近,越来越多的学者开始考虑金融保险中的隐马尔科夫模型中投资与再保险问题,例如B(a|¨)uerle and Rieder[17],Elliott and van der Hoek[49],Elliott et al.[51],Lakner[97],Nagai and Runggaldier[126],Pham andQuenez[132],Rieder and B(a|¨)uerle[141],Sass[146],Sass and Haussmann[147]等。Lakner[97]利用鞅方法研究了当金融市场中股票回报率无法观测情况下的最优投资与消费问题并且对对数效用与幂效用得到了显式表达。Lakner[98]利用同样的方法解决了股票回报率为线性扩散过程时相应的问题。B(a|¨)uerle andRieder[17],Sass and Haussmann[147]考虑了当金融市场中股票的回报率受一投资者无法观测的马氏链调节情况下最大化期望效用的投资组合问题。B(a|¨)uerle and Rieder[17]则考虑了带跳金融市场中无法观测跳的强度下的最大化期望效用的投资组合问题。Nagai and Runggaldier[126]则通过偏微分方程理论研究了隐马尔科夫模型中最大化期望效用的投资组合问题。
纵观以上文章,我们发现它们都只是考虑隐马尔科夫模型中的最优投资问题,并没有涉及再保险问题。因此在第9章中,我们将考虑保险公司在局部市场信息下,如何采取再保险与投资策略来最大化终值指数效用。此处局部信息是指,对于股票市场,保险公局只能知道股票价格的信息,而相应的股票的回报率是受一无法观测的马氏链控制,即此处的市场为隐马尔科夫金融市场。在此章中,保险公司的盈余过程由带漂移的布朗运动来描述,而所考虑的隐马尔科夫金融市场是一个具有多个风险资产的市场。通过建立分离原理我们将原有的局部信息问题转化成完全信息问题。利用动态规划原则以及Girsanov变换方法,我们得到最优值函数是一线性抛物偏微分方程的唯一粘性解。
Compared with the classical risk model or finance model, the Markov-modulated model or Markovian regime-switching model seems to provide a better fit to the reality data of insurance and finance. In risk theory, the Markov-modulated risk model can capture the feature that insurance policies may 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 process in different weather conditions will be very different. Therefore the insurance policies of insurance companies, such as premium rate, will be different in different weather conditions. In finance theory, the celebrated Black-Scholes-Merton financial market is based on a geometric Brownian motion to capture the price dynamics of the underlying security. However, numerous empirical studies reveal that this assumption for assets price dynamics cannot provide realistic description for some important empirical behavior of financial returns such as the heavy-tailedness of the return's distribution and the time-varying volatility of return. Hardy [75] shows that the Markovian regime-switching model provides a significantly better fit to the data than do other popular models. Therefore, the class of Markov-modulated or Markovian regime-switching model becomes more and more important in insurance and finance.
On the basis of these background, my doctoral dissertation is mainly devoted to considering the applications of Markov-modulated or Markovian regime-switching model in insurance and finance. The dissertation is organized as follows.
A brief overview of the history of Markov-modulated or Markovian regime-switching model in risk theory and finance theory, as well as the main contents of this dissertation, are given in the first chapter.
The second chapter is the theoretical foundation of this dissertation. In this chapter, we review some basic definitions and results on double martingales, continuous time Markov chain and Markov-modulated Lévy processes. The main contribution of this chapter is that we introduce the Markov chain following the framework of Elliott et al. [51] and characterize the Markov chain by associating it with a marked point process. Based on the marked point process characterization of Markov chain, we introduce the j~(th) Markov jump martingale which plays a major rule in completing the Markovian regime-switching market in Chapter 4. Besides, we also derive some new results, such as the integral representation of the function of Markov chain and the It(o|^) differential rule for the generalized Markov-modulated stochastic process, which are very useful in characterizing the measure change of Markov-modulated Brownian motion in Chapter 4 and proving the verification theorem for the optimization problems in the sequent chapters.
The remaining chapters are devoted to solving the following five problems of the Markov-modulated or Markovian regime-switching model in insurance and finance.
Gerber-Shiu discounted penalty function of Markov-modulated compound Poisson risk model. Risk theory received a substantial boost with the article of Gerber and Shiu [66] in 1998, in which the Gerber-Shiu discounted penalty function was introduced. Since the basic actuarial variables such as the time of ruin, the surplus immediately prior to ruin, the deficit at ruin, are embedded in the Gerber-Shiu discounted penalty function, a large number of papers in the ruin literature are devoted to obtaining closed forms of Gerber-Shiu discounted penalty functions in a variety of risk models, see Wang and Wu [163] for the perturbed compound Poisson risk process with constant interest, Zhang et al. [178] for classical risk model with a two-step premium rate, Landriault and Willmot [99], Willmot [165], Willmot and Dickson [166] for the renewal risk model, Garrido and Morales [63], Morales [122], Morales and Olivares [123] for the Lévy risk model. Although the Markov-modulated risk model was first introduced by Janssen [88] and later treated by Janssen and Reinhard [89], Reinhard [140], Asmussen [5, 7], B(a|¨)uerle [14], Wu [169] ,Wu and Wei [167], Ng and Yang [128, 129] and so on, Albrecher and Boxma [2] were the first to consider the Gerber-Shiu discounted penalty function for the Markov-dependent risk model. Later Ng and Yang [129] also considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. However, they only gave a system of integro-differential equations for Gerber-Shiu functions and did not discuss the solution of this system of integro-differential equations. Inspired by the work of Albrecher and Boxma [2] and Ng and Yang [129], we study the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model and obtain closed forms of Gerber-Shiu discounted penalty functions for this model, see Chapter 3 of my doctoral dissertation. Later Lu and Tsai [110] considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model perturbed by diffusion.
Chapter 3 is devoted to obtaining closed forms of the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. We first present a system of integro-differential equations of the Gerber-Shiu discounted penalty functions for arbitrary premium rate. By using Laplace transform to solve the integro-differential equations, we obtain explicit expressions of the Laplace transforms of Gerber-Shiu discounted penalty functions. Closed forms of the discounted joint density function of surplus prior to and after ruin for the initial surplus 0 are obtained. Besides, we derive explicit formulas of Gerber-Shiu discounted penalty functions for the K_n-family claim size distributions in the two-state case through introducing the operator T_r. Finally, numerical examples are presented to illustrate our results.
Completing the Markovian regime-switching market via double martingales. Completeness of the financial market is an important concept in the finance literature and the theory of the complete market is more mature than that of the incomplete market. Therefore, if we can find a method to complete the incomplete market, we can use the results of the complete market to solve the problem in the original incomplete market. Karatzas et al. [95] introduced a way to complete the incomplete market by adding the fictitious stocks for the original incomplete market. By using the results of utility maximization problem in the complete market and the relationship between the complete market and the original incomplete market, Karatzas et al. [95] solved the utility maximization problem in the original incomplete market. The Markovian regime-switching market consists of one bond and one stock, see Chapter 4 for the description of this market. It is well-known that the Markovian regime-switching market is incomplete (see Elliott and Swishchuk [48]). So we want to find a method to complete the Markovian regime-switching market and study the portfolio optimization in the completed Markovian regime-switching market. The idea of completing the Markovian regime-switching is inspired by Corcuera et al. [34], Corcuera et al. [35], Niu [130] and Guo [71]. By adding the power-jump assets (related to the suitably compensated power-jump processes of the underlying Lévy process), Corcuera et al. [34] completed the geometric Lévy market, while, Corcuera et al. [35] considered the optimal investment problems in the completed geometric Lévy market with the new power-jump assets. Niu [130] studied the optimal investment problems in a simple Lévy market which was completed by adding the power-jump assets introduced by Corcuera et al. [34]. Guo [71] introduced the change-of-state contract to complete the Markovian regime-switching market. However, this method is not good enough for the solution of optimal investment problems. Therefore we need to find some other method to complete the market and consider the optimal investment problems.
In Chapter 4 and Chapter 5, we find two new methods to complete the Markovian regime-switching market via the theory of double martingales and marked point process. The problem of portfolio optimization for maximizing expected utility in the completed Markovian regime-switching market is well solved.
In Chapter 4, we first give a characterization of all structure-preserving equivalent martingale measure, under which the Markov-modulated Brownian motion remains a Markov-modulated Brownian motion but with changed parameters. Then by using the martingale representation technique of double martin- gales introduced by Elliott [44], we complete the Markovian regime-switching market by adding the j~(th) Markovian jump assets. We prove that the enlarged Markovian regime-switching market(i.e. the Markovian regime-switching market completed by adding the j~(th) Markovian jump assets.) is complete and the explicit hedging policy for a non-negative square-integrable contingent in the enlarged Markovian regime-switching market is also obtained. We also prove that the enlarged Markovian regime-switching market is arbitrage-free and the unique equivalent martingale measure is determined by the explicit expression of likelihood ratio process.
In Chapter 5, we make a minor modification to the method of completing the Markovian regime-switching market in Chapter 4. One potential problem of the j~(th) Markovian jump assets introduced in Chapter 4 is that it can take negative values with positive probability. To exclude this potential problem, we consider a set of j~(th) Markovian geometric jump assets, which always take non-negative values. We prove that the Markovian regime-switching market enlarged by a set of j~(th) Markovian geometric jump assets is complete and arbitrage-free. The portfolio selection problem in the enlarged market in the case of a power utility and a logarithmic utility is well solved. In the case of logarithmic utility we adopt the direct differentiation approach to derive the closed-form solution of the optimal portfolio strategy, while for the power utility, we adopt the dynamic programming approach to derive the optimal portfolio strategies. Closed-form solutions for the optimal portfolio strategies and the value function are obtained in both cases. Besides, we also give a Feynman-Kac representation of the value function in both utility cases.
Maximizing the expected utility with proportional reinsurance and investment in the extended Markovian regime-switching market with multiple risky assets under no-shorting constraint. Expected utility, as an important objective function in the financial and actuarial literature, has attracted a great deal of interest, see Bai and Guo [10, 11], Brendle [23], Browne [25], Cvitanic and Karatzas [37], Karatzas et al. [95], Lakner [97], Pham and Quenez [132], Yang and Zhang [172] and so on. Browne [25] considered the optimal investment policies for maximizing exponential utility and minimizing the probability of ruin. Bai and Guo [10] extended the result of Browne [25] to the case of the financial market with multiple risky asset and no-shorting constraint. Bai and Guo [11] also considered the utility maximization in the case of partial information. Yang and Zhang [172] studied the optimal investment policies of an insurer with jump-diffusion risk process. The explicit expressions for the optimal strategy and the value function are given.
In contrast to above papers, we consider the problem of maximizing the expected utility in the Markov-modulated case, i.e., the surplus of the insurance company and the prices of the financial market are modulated by a continuous time Markov chain. The financial market considered in Chapter 6 and Chapter 7 is the extended Markovian regime-switching market. There are two main differences between the extended Markovian regime-switching market and the Markovian regime-switching market considered in Chapter 4 and Chapter 5: firstly, the extended Markovian regime-switching market consists of one bond and d stocks while the Markovian regime-switching market considered in Chapter 4 and Chapter 5 only consists of one bond and one stock; secondly, in the extended Markovian regime-switching market the bank interest rates, stocks' appreciation rates, and volatility rates not only are modulated by a continuous Markov chain, but also depend on time t.
In Chapter 6, we adopt the direct differentiation approach and dynamic programming approach to solve the portfolio selection problems in the extended Markovian regime-switching market for the logarithmic and power utility respectively. Closed-form solutions for the optimal portfolio strategies and the Feynman-Kac representation of value functions in both logarithmic and power utility with no shorting constraint or not are obtained.
In Chapter 7, we focus on the optimal proportional reinsurance and investment in the extended Markovian regime-switching market. The main differences from the classical optimal proportional reinsurance and investment problems (see Browne [25]) are: firstly, the surplus of the insurance company is modeled by a Markov-modulated Brownian motion; secondly, the financial market is the extended Markovian regime-switching market; thirdly, no shorting constraint is considered in our problem. We adopt the dynamic programming approach to obtain the optimal proportional reinsurance and investment strategy for maximizing exponential utility function from terminal wealth under the no-shorting constraint. We first present the Hamilton-Jacobi-Bellman (HJB) equation satisfied by the optimal value function and then construct the solution of the HJB equation by the usual separable variable approach. At last, we verify that the solution of the HJB equation is indeed the optimal value function. Closed-form of the optimal proportional reinsurance and investment strategies and the Feynman-Kac representation of the optimal value function are obtained. Besides, we find that the first part of Bai and Guo [10] is a special case of our results. We verify the result of Bai and Guo [10] through our model.
Mean-Variance problem in the Markov-switching jump-diffusion market. Compared with the utility maximization problems, mean-variance problem enables an investor to seek the highest return after specifying his/her acceptable risk level that is quantified by the variance of the return. The mean-variance problem was originally proposed by Markowitz [113, 114] for portfolio construction in a single period. Now the mean-variance approach has become the foundation of modern finance theory and has inspired numerous extensions and applications. The continuous time mean-variance problem in the geometric Brownian motion market has been considered by many authors, see Bielecki et al. [20], Li et al. [102], Lim [103], Lim and Zhou [104], Xiong and Zhou [170], Zhou and Li [181], and so on. However, the papers on the mean-variance problem in the Markovian regime-switching market are quite few. So far as we know, there are only three papers on the mean-variance problem in the Markovian regime-switching market, see Zhou and Yin [182], Yin and Zhou [174] and Chen et al. [31]. The above three papers only consider mean-variance problem in the case of the Markovian regime-switching market without jump. So in Chapter 8, we consider mean-variance problem in the Markov-switching jump-diffusion market.
In Chapter 8, we first present the description of the Markov-switching jump- diffusion market and reduce the mean-variance problem to a stochastic linear-quadratic (LQ) problem by using the method of Zhou and Li [181]. The feasibility of the mean-variance problem is discussed and some necessary and sufficient conditions for the feasibility in Markov-switching market are given. Through dynamic programming technique, we solve the stochastic LQ problem and obtain the closed form of the value function. Prom the relationship between the stochastic LQ problem and the original mean-variance problem and the well-known Lagrange duality theorem, we obtain efficient portfolio and efficient frontier in a closed form. We also obtain the minimum variance, i.e. the minimum possible terminal variance, along with the portfolio that attains the minimum variance. Besides, we also prove that the mutual fund theorem in the Markov-switching market still holds.
Optimal reinsurance and investment in a hidden Markov model. Recently, more and more papers consider the hidden Markov model in insurance and finance, see Bauerle and Rieder [17], Elliott and van der Hoek [49], Elliott et al. [51], Lakner [97], Nagai and Runggaldier [126], Pham and Quenez [132], Rieder and Bauerle [141], Sass [146], Sass and Haussmann [147], and so on. Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Bauerle and Rieder [17], Sass and Haussmann [147] considered the problem of maximizing the expected utility of the terminal wealth in the case that the drift rate of the stock is Markov-modulated and can not be observed by the investor. Bauerle and Rieder [17] studied the hidden Markov jump intensity model. Using a classical result from filter theory, the hidden problem was reduced to the complete observation problem. Nagai and Runggaldier [126] considered the utility maximization for market models with hidden markov factors through PDE approach.
Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Pham and Quenez [132] addressed the maximization problem of expected utility from terminal wealth in a financial market where price process of risky assets follows a stochastic volatility model and only the vector of stock prices are observed by the investors.
Based on these papers, in Chapter 9, we consider an optimal reinsurance and investment problems in a multiple risky assets market with appreciation rate driven by a hidden Markov chain. The surplus of the insurance company is modeled by a Brownian motion with drift and the objective function is the expected exponential utility. By using the filter theory, we establish the separation principle and reduce the problem to the completely observed case. Through the dynamic programming approach and the Girsanov change of measure, we characterize the value function as the unique viscosity solution of a linear parabolic partial differential equation and obtain the Feynman-Kac representation of the value function.
引文
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