两类风险模型的最优分红控制策略

发布时间：2018-03-06 14:23

本文选题：随机分析　切入点：随机过程　出处：《兰州理工大学》2014年硕士论文　论文类型：学位论文

【摘要】：数学工具在金融工程中获得了越来越多的关注,尤其是Gerber,H.U等人将鞅的理论和方法应用到风险理论中,使得该学科得到了迅速的发展,定价理论更是成为了资产组合理论、资本资产定价模型之后获得诺贝尔经济学奖的重要理论。随着金融与保险市场发展,保险公司不再仅满足于求得破产概率,破产时间等几个精算量,转而寻求某种措施使得风险最小,或者收益达到最大。这些都属于金融保险中的最优控制问题。过去几十年里,通过随机控制的理论,尤其是通过Hamilton-Jacobi-Bellman (HJB)方程的方法更使得该领域进展迅速,并开创了风险理论和随机控制理论相结合的先例。然而,对风险模型的最优控制的研究,大多数工作集中在其数值解上面。但若追求数值解的精度则势必在其求解速度上有所牺牲,尤其很多工作所采用的蒙特卡洛方法在面对瞬息万变的金融环境时显得力不从心。本文在经典模型基础上通过添加符合实际的因素,对两种最优分红模型及其明确的最优控制进行研究并得到了控制策略和价值函数的具体表达式。本文主要分为两个部分,每个部分建立了一种与实际更贴近的风险模型及其相关的最优分红策略。 (1)首先在本文第一部分中将经典模型中的复合Posioon过程通过布朗运动来近似,将经典模型拓展为了一个基于连续时间盈余过程的风险模型,同时在模型中引入了三种控制策略并讨论了相应的最优分红问题。三种控制策略分别为当前时刻的分红策略,风险资产的投资数额和比例再保险策略。在求出最优的价值函数以及最优控制策略的具体形式之前,给出了风险模型的性质,这些性质不依赖于解的具体表达式。这部分内容在本文第二章进行讨论。 (2)第二部分讨论保费率低于最高分红率限制下风险模型与最优分红策略问题。模型假设索赔大小的分布是指数分布,除了引入必要的分红策略外,本文在价值函数中引入了随时间变化的参量,使得最终所得到的控制策略以及价值函数不止局限于当前时刻而是得到了任意时刻的控制策略和价值函数,这部分内容在本文第三章进行讨论。每章里又分为小节,依次顺序为介绍模型、HJB方程导出、求解最优策略,数值模拟以及模型小结。本论文一个非常重要的特色就是对于所有的最优问题都给出了非常明确的最优解。
[Abstract]:The mathematical tools to get more and more attention in financial engineering, especially Gerber, H.U et al. The theory and method of application of martingale theory into the risk theory, the subject has been rapid development, pricing theory has become an important theory, portfolio theory, capital asset pricing model, won the Nobel prize in economics with. Finance and insurance market, the insurance company not only meet in the ruin probability, ruin time and several actuarial variables, to seek some measures to make the risk minimum, or to achieve the largest profit. These are the optimal control problems in finance and insurance. In the past few decades, the stochastic control theory, especially the Hamilton-Jacobi-Bellman (HJB) equation method makes it more rapid progress, and created the risk theory and stochastic control theory combined with the precedent. However, the risk model. Study on the optimal control of the most concentrated in the numerical solutions of the above work. But if the pursuit of the accuracy of the numerical solution is bound in the solving speed of sacrifice, especially by using Monte Carlo method a lot of work in the face of rapidly changing financial environment appeared to be inadequate. In this paper, on the basis of the classical model by adding factors with the actual to conduct research and specific expression of control strategy and the value function of the control of two kinds of optimal dividend model and explicit optimal. This paper is divided into two parts, each part of the establishment of a more close to the actual risk model and the optimal dividend strategy.
(1) the first compound Posioon process in the first part, the classical model by Brown motion approximation, the classical model is extended to a risk model based on continuous time surplus process at the same time, the model introduces the three kinds of control strategies and discuss the optimal dividend problems. Three kinds of control strategies for the current when the dividend strategy, investment amount and reinsurance strategy of risk assets. Before calculating the optimal value function and the specific form of the optimal control strategy, given the nature of the risk model, these properties do not depend on the expression of the solution. The contents of this part are discussed in the second chapter of this article.
(2) the second part discusses the premium rate is lower than the highest dividend rate limit risk model and the optimal dividend strategy. Distribution model assumes that the claim size is exponential distribution, in addition to the introduction of the necessary dividend strategy, this paper introduces the parameter changes over time in the value function, the control strategy of the resultant function and the value of more than but the time is limited to the current control strategy and the value function at any time, the contents of this part are discussed in the third chapter of this article.
Each chapter is divided into sub sections, which are sequentially introduced as the introduction model, the HJB equation is derived, and the best strategy, numerical simulation and model summary are presented. A very important feature of this paper is a very clear optimal solution for all the optimal problems.

【学位授予单位】：兰州理工大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O211.67;F840

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