马氏调节风险模型的若干破产问题研究
发布时间：2018-10-23 15:55
【摘要】：经过众多精算学者的不断努力，作为现代风险管控三大技术之一的破产理论在最近几十年取得了深刻的发展。传统的风险模型已然不能较为有效地刻画当今各金融行业的实际运作模式，各种更为深刻的风险建模方法及风险度量技术随之应运而生。在各种推广的风险模型的基础上加入对多种经济因素的考量业已成为当今破产理论的研究热点之一。本文主要基于很广泛的一类半马尔科夫相依风险模型的基础，着重考察常数利息力以及重尾理赔对破产概率的影响。 在第二章，本文主要考虑Albrecher and Boxma(2005)中所介绍的一类齐次半马尔科夫相依风险模型（HSMRM），并加入对常数利息力的考量。通过基本的概率分析方法，推导出相关生存概率所满足的矩阵积分-微分方程，并在此基础上得到了形式更为简洁的矩阵积分方程。 在第三章，本文主要考虑了第二章中的一种简化情形，即两状态的SMRM。当“系统”运行至状态1时，理赔额随机变量将服从参数为β的指数分布，下一次理赔时间间隔随机变量服从参数为λ_1的指数分布；当“系统”运行至状态2时，理赔额服从任意的重尾分布F_2，下一次的理赔时间间隔服从参数为λ_2的指数分布。本章首先利用拉普拉斯变换，将破产概率所满足的积分-微分方程组转化为二阶变系数线性常微分方程，再由常微分方程求解理论得到破产概率拉氏变换的级数形式解，最后结合the Karamata Tauberian Theorem以及the Heaviside OperationalPrinciple得到最终破产概率所满足的渐进表达式。本章结果表明破产概率的渐进行为只依赖于状态2伴随的重尾理赔额分布F_2，，而与状态1伴随的指数理赔无关。本章的结果可以推广至一类n (n≥2)个状态的情形，此时任一状态对应的破产概率的拉氏变换将满足某一n阶变系数线性常微分方程，整个求解、分析过程将变得更为繁琐。 在第四章，本文通过对第二章中所述模型的初始状态分布以及状态间的转移概率加以假设，对一类平稳的SMRM加以研究。利用这类模型中理赔时间间隔随机变量与理赔额随机变量间的条件独立性，结合基本的概率分析方法，论证发现这种平稳风险模型本质上是一类更新风险模型。进而利用Hao and Tang(2008)中的相关结果，直接推导出该模型下有限时间破产概率的一致渐进行为，而如果对理赔时间间隔、理赔额随机变量再加以某些宽松约束，也可以推导出最终破产概率的一致渐进表达式。
[Abstract]:Through the continuous efforts of many actuaries, bankruptcy theory, as one of the three modern risk management techniques, has made a profound development in recent decades. The traditional risk model has been unable to describe the actual operation mode of various financial industries effectively, and various more profound risk modeling methods and risk measurement techniques have come into being. On the basis of various generalized risk models, the consideration of various economic factors has become one of the hot topics in bankruptcy theory. In this paper, based on a widely used semi-Markov dependent risk model, the effects of constant interest force and heavy-tailed claim on ruin probability are investigated. In chapter 2, we consider a homogeneous semi-Markov dependent risk model, (HSMRM), which is introduced in Albrecher and Boxma (2005, and consider the constant interest force. By using the basic probability analysis method, the matrix integro-differential equation satisfied by the correlation survival probability is derived, and on this basis, a more concise matrix integral equation is obtained. In Chapter 3, we mainly consider a simplified case in Chapter 2, that is, two-state SMRM.. When the "system" runs to state 1, the claim amount random variable takes the exponential distribution from the parameter 尾, the next claim interval random variable from the exponential distribution of the parameter 位 _ 1, and when the "system" runs to state 2, The amount of claim is distributed from any heavy-tailed distribution F _ 2, and the next claim interval is from an exponential distribution of 位 _ 2 parameters. In this chapter, the integro-differential equations satisfied by ruin probability are transformed into second-order linear ordinary differential equations with variable coefficients by means of Laplace transformation, and then the series solution of Laplace transformation of ruin probability is obtained by solving theory of ordinary differential equation. Finally, combined with the Karamata Tauberian Theorem and the Heaviside OperationalPrinciple, the asymptotic expression of the final ruin probability is obtained. The results show that the asymptotic behavior of the ruin probability depends only on the distribution of heavy-tailed claims with state 2, but not on the exponential claims accompanied by state 1. The results in this chapter can be extended to a class of n (n 鈮
本文编号：2289710
[Abstract]:Through the continuous efforts of many actuaries, bankruptcy theory, as one of the three modern risk management techniques, has made a profound development in recent decades. The traditional risk model has been unable to describe the actual operation mode of various financial industries effectively, and various more profound risk modeling methods and risk measurement techniques have come into being. On the basis of various generalized risk models, the consideration of various economic factors has become one of the hot topics in bankruptcy theory. In this paper, based on a widely used semi-Markov dependent risk model, the effects of constant interest force and heavy-tailed claim on ruin probability are investigated. In chapter 2, we consider a homogeneous semi-Markov dependent risk model, (HSMRM), which is introduced in Albrecher and Boxma (2005, and consider the constant interest force. By using the basic probability analysis method, the matrix integro-differential equation satisfied by the correlation survival probability is derived, and on this basis, a more concise matrix integral equation is obtained. In Chapter 3, we mainly consider a simplified case in Chapter 2, that is, two-state SMRM.. When the "system" runs to state 1, the claim amount random variable takes the exponential distribution from the parameter 尾, the next claim interval random variable from the exponential distribution of the parameter 位 _ 1, and when the "system" runs to state 2, The amount of claim is distributed from any heavy-tailed distribution F _ 2, and the next claim interval is from an exponential distribution of 位 _ 2 parameters. In this chapter, the integro-differential equations satisfied by ruin probability are transformed into second-order linear ordinary differential equations with variable coefficients by means of Laplace transformation, and then the series solution of Laplace transformation of ruin probability is obtained by solving theory of ordinary differential equation. Finally, combined with the Karamata Tauberian Theorem and the Heaviside OperationalPrinciple, the asymptotic expression of the final ruin probability is obtained. The results show that the asymptotic behavior of the ruin probability depends only on the distribution of heavy-tailed claims with state 2, but not on the exponential claims accompanied by state 1. The results in this chapter can be extended to a class of n (n 鈮
本文编号：2289710
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