跳扩散模型在风险理论中的应用
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摘要
集体风险理论主要用来研究保险公司的风险行为,百年来一直备受关注,是精算数学中重要的一支。其经典模型由Lundberg在1903的论文(Lundberg(1903))中引进,而后经过Harald Cramer等人的努力,使得模型在数学上的定义更为严谨,被大多数人接受。经典模型,同时也被称为Cramer-Lundberg风险模型,将所讨论的公司模型刻画为复合Poisson过程,在很多方面得到了应用和发展。Gerber(1970)提出的带干扰的复合Poisson模型和Andersen(1957)提出的更新模型,也就是Sparre Anderson模型就是其中比较有名的成功的例子。伴随模型的发展和丰富,多种多样的理论,方法和函数被引入风险理论的研究中。比如,更新理论,Winener-Hoff方程,It(?)公式,逐段决定马氏过程还有鞅方法都是风险理论及其相关领域中比较常用的方法。Gerber and Shiu(1997,1998a)在古典模型中引入Gerber-Shiu罚金函数,使得在精算中最重要三个变量,破产时间,破产赤字,破产前盈余,完美的统一在一起,之后由Tsai and Willmot(2002)发展到带干扰的模型中。
与此同时,带分红策略下的风险模型也是风险理论中备受关注的问题。当然这部分也是由于模型本身所具有的现实意义所决定的。分红,即是指将公司的部分所得作为红利分发给公司所有者或股份参与者。自然,对于这些受益人而言,他们不仅关心公司目前的经济状态,更关心的是采取怎么样的分红策略才能使自己的收益以一定的折现率折现后尽可能的大,即所谓的最优分红问题。根据不同的客户要求,或者说在不同的分红限制下,最好的分红策略自然是不同的。现在比较公认的好策略有两种,一种是带分红壁的分红策略,另外一种是门槛分红策略,他们已经被证明是在相应的限制下最优的。这两种分红策略下的风险模型将在第2章和第3章讨论。另外,近来一种称为多门槛的分红策略也被引入风险模型中,并且引起了很多人的注意。多门槛策略下的马氏调节过程也在本论文的第4章进行了讨论。
基于上述的背景,我的博士毕业论文主要是致力于对一些分红模型中的Gerber-Shiu罚金函数以及期望折现分红函数进行研究。
第1章主要简介了本论文的研究背景及本毕业论文的结构和内容。
第2,3章主要研究所涉函数的边界可微性,证明了所讨论的函数在分红边界是一阶可微的。第2章中,我们讨论了在门槛分红策略下带干扰的复合Poisson模型,对这个模型的期望折现分红函数和Gerber-Shiu罚金函数进行了研究,证明了他们在分红边界的可微性,并借助积分微分方程的常规处理方法,给出了他们的表达式。
在第3章中,受Avanzi et al.(2007)中所讨论的模型的启发,我们研究了带壁分红策略下,受布朗运动干扰的对偶模型,并计算了这个模型的期望折现分红函数。以证明函数在分红边界的可微性为第一目的,然后用通常的方法对这个函数进行了求解。
在第4章中,我们先研究了一类更新的跳扩散模型,并将结论推广到了受马氏过程调节的跳扩散模型,其中跳的分布为相位分布。受Asmussen(1995)和Bladt(2005)的启发,借助于随机流的分析思路,讨论了过程的首出时问题,并用矩阵的形式对破产相关的变量进行了表示。比如破产时间,破产前盈余,破产赤字三者的联合分布。由于涉及矩阵形式的方程,给出了所讨论方程的可解性,以及解的唯一性。之后应用前面两章的结论,我们给出了这个模型在单门槛分红和多门槛分红策略下的期望分红函数。
Collective risk theory, considering a model of the risk business of an insurance company, has been studied for more than a century and always been a vital part of actuarial mathematics. It is well known that the classical risk model was firstly introduced by Lundberg (1903), and was later developed and made mathematically rigorous by Harald Cramer in Cramer (1930). This model, also known as the Cramér-Lundberg risk model, models the surplus of an insurance business as a compound Poisson process and has since then been studied and extended in various ways. The perturbed compound Poisson risk process of Gerber (1970) and the Sparre Andersen risk process of Andersen (1957) are the most famous extensions. Together with the generalizations of the model, there are diversified approaches and functionals introduced to study risk process. Renewal theory, Wiener-Hoff equation, It(o|^) formula, piecewise deterministic Markov processes and the martingale methods are some of the most commonly used approaches in risk theory and other disciplines. See Rolski et al. (1999) and references therein. And three most important actuarial variables, the time of ruin, the deficit at ruin, the surplus immediately before ruin, are also captured by introduction of a so-called Gerber-Shiu discounted penalty function in Gerber and Shiu (1997, 1998a) and Tsai and Willmot (2002).
On the other hand, due to its practical importance, the issue of dividend strategies becomes an increasingly popular branch of risk theory. Dividends are payments made to stockholders from a firm's earnings, it is desirable to find a fixed rule which produces largest possible expected sum of discounted dividend, and that is the optimal dividend problem. Among all the dividend policies associated with different criterions, the barrier dividend strategy and the threshold dividend strategy are of particular interest, and in the literature of classical risk model, they are the "best" under their corresponding constraints. Therefore, we consider risk models with the presence of these dividend strategies in Chapter 2 and Chapter 3. And more recently, there is another hot topic of so-called multi-layer threshold dividend model which becomes popular, a Markov-modulated risk model with this dividend strategy is studied in Chapter 4. On the basis of these background, this thesis is mainly devote to the analysis of the Gerber-Shiu discounted penalty function(also called Gerber-Shiu function later) and the expected discounted dividend function of some risk model.
Firstly, we shortly review the history of risk theory, the most commonly used methods, functions and models are presented in the first Chapter, as well as those now classical works. And the organization of this thesis is also given in this chapter. Then, the main body of the thesis starts.
The next two chapters are mainly concerned on the proof of differentiable of the functions at the dividend barriers. In chapter 2, we considered the perturbed compound Poisson risk model with a threshold dividend strategy. The expected discounted dividend function and the Gerber-Shiu discounted penalty function were studied. Inspired by Wang and Wu (2000) and Wang (2001), we proved the boundary conditions, following the common way of solving integro-differential equations, and expressed the function concerned in terms of some other functions.
In chapter 3, motivated by the model in Avanzi et al. (2007), we considered a dual model with perturbations of Brownian motion and barrier dividend. The expected discounted dividend function was studied. And we also aimed at finding its boundary conditions, and then expressed it in the usual way.
And in chapter 4, a perturbed renewal jump-diffusion risk model is studied, the conclusions are extended to a Markov-modulated model with the claim sizes specified to be in the class of phase type distributions. Motivated by Asmussen (1995) and Bladt (2005), following the fluid flow method, we studied some passage time of the process, and expressed the ruin related quantities in matrix form, such as the joint distribution of the ruin time, the surplus before ruin and the deficit at ruin. And we also success in providing the necessary steps that should be taken to ensure the solvability of matrix equation considered, as well as the uniqueness of the solution. And then, by applying the conclusions from previous chapters, we also studied this process with perturbation of threshold dividend and multi-layer dividend strategy.
与此同时,带分红策略下的风险模型也是风险理论中备受关注的问题。当然这部分也是由于模型本身所具有的现实意义所决定的。分红,即是指将公司的部分所得作为红利分发给公司所有者或股份参与者。自然,对于这些受益人而言,他们不仅关心公司目前的经济状态,更关心的是采取怎么样的分红策略才能使自己的收益以一定的折现率折现后尽可能的大,即所谓的最优分红问题。根据不同的客户要求,或者说在不同的分红限制下,最好的分红策略自然是不同的。现在比较公认的好策略有两种,一种是带分红壁的分红策略,另外一种是门槛分红策略,他们已经被证明是在相应的限制下最优的。这两种分红策略下的风险模型将在第2章和第3章讨论。另外,近来一种称为多门槛的分红策略也被引入风险模型中,并且引起了很多人的注意。多门槛策略下的马氏调节过程也在本论文的第4章进行了讨论。
基于上述的背景,我的博士毕业论文主要是致力于对一些分红模型中的Gerber-Shiu罚金函数以及期望折现分红函数进行研究。
第1章主要简介了本论文的研究背景及本毕业论文的结构和内容。
第2,3章主要研究所涉函数的边界可微性,证明了所讨论的函数在分红边界是一阶可微的。第2章中,我们讨论了在门槛分红策略下带干扰的复合Poisson模型,对这个模型的期望折现分红函数和Gerber-Shiu罚金函数进行了研究,证明了他们在分红边界的可微性,并借助积分微分方程的常规处理方法,给出了他们的表达式。
在第3章中,受Avanzi et al.(2007)中所讨论的模型的启发,我们研究了带壁分红策略下,受布朗运动干扰的对偶模型,并计算了这个模型的期望折现分红函数。以证明函数在分红边界的可微性为第一目的,然后用通常的方法对这个函数进行了求解。
在第4章中,我们先研究了一类更新的跳扩散模型,并将结论推广到了受马氏过程调节的跳扩散模型,其中跳的分布为相位分布。受Asmussen(1995)和Bladt(2005)的启发,借助于随机流的分析思路,讨论了过程的首出时问题,并用矩阵的形式对破产相关的变量进行了表示。比如破产时间,破产前盈余,破产赤字三者的联合分布。由于涉及矩阵形式的方程,给出了所讨论方程的可解性,以及解的唯一性。之后应用前面两章的结论,我们给出了这个模型在单门槛分红和多门槛分红策略下的期望分红函数。
Collective risk theory, considering a model of the risk business of an insurance company, has been studied for more than a century and always been a vital part of actuarial mathematics. It is well known that the classical risk model was firstly introduced by Lundberg (1903), and was later developed and made mathematically rigorous by Harald Cramer in Cramer (1930). This model, also known as the Cramér-Lundberg risk model, models the surplus of an insurance business as a compound Poisson process and has since then been studied and extended in various ways. The perturbed compound Poisson risk process of Gerber (1970) and the Sparre Andersen risk process of Andersen (1957) are the most famous extensions. Together with the generalizations of the model, there are diversified approaches and functionals introduced to study risk process. Renewal theory, Wiener-Hoff equation, It(o|^) formula, piecewise deterministic Markov processes and the martingale methods are some of the most commonly used approaches in risk theory and other disciplines. See Rolski et al. (1999) and references therein. And three most important actuarial variables, the time of ruin, the deficit at ruin, the surplus immediately before ruin, are also captured by introduction of a so-called Gerber-Shiu discounted penalty function in Gerber and Shiu (1997, 1998a) and Tsai and Willmot (2002).
On the other hand, due to its practical importance, the issue of dividend strategies becomes an increasingly popular branch of risk theory. Dividends are payments made to stockholders from a firm's earnings, it is desirable to find a fixed rule which produces largest possible expected sum of discounted dividend, and that is the optimal dividend problem. Among all the dividend policies associated with different criterions, the barrier dividend strategy and the threshold dividend strategy are of particular interest, and in the literature of classical risk model, they are the "best" under their corresponding constraints. Therefore, we consider risk models with the presence of these dividend strategies in Chapter 2 and Chapter 3. And more recently, there is another hot topic of so-called multi-layer threshold dividend model which becomes popular, a Markov-modulated risk model with this dividend strategy is studied in Chapter 4. On the basis of these background, this thesis is mainly devote to the analysis of the Gerber-Shiu discounted penalty function(also called Gerber-Shiu function later) and the expected discounted dividend function of some risk model.
Firstly, we shortly review the history of risk theory, the most commonly used methods, functions and models are presented in the first Chapter, as well as those now classical works. And the organization of this thesis is also given in this chapter. Then, the main body of the thesis starts.
The next two chapters are mainly concerned on the proof of differentiable of the functions at the dividend barriers. In chapter 2, we considered the perturbed compound Poisson risk model with a threshold dividend strategy. The expected discounted dividend function and the Gerber-Shiu discounted penalty function were studied. Inspired by Wang and Wu (2000) and Wang (2001), we proved the boundary conditions, following the common way of solving integro-differential equations, and expressed the function concerned in terms of some other functions.
In chapter 3, motivated by the model in Avanzi et al. (2007), we considered a dual model with perturbations of Brownian motion and barrier dividend. The expected discounted dividend function was studied. And we also aimed at finding its boundary conditions, and then expressed it in the usual way.
And in chapter 4, a perturbed renewal jump-diffusion risk model is studied, the conclusions are extended to a Markov-modulated model with the claim sizes specified to be in the class of phase type distributions. Motivated by Asmussen (1995) and Bladt (2005), following the fluid flow method, we studied some passage time of the process, and expressed the ruin related quantities in matrix form, such as the joint distribution of the ruin time, the surplus before ruin and the deficit at ruin. And we also success in providing the necessary steps that should be taken to ensure the solvability of matrix equation considered, as well as the uniqueness of the solution. And then, by applying the conclusions from previous chapters, we also studied this process with perturbation of threshold dividend and multi-layer dividend strategy.
引文
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