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保费收取额与收取时间间隔是FGM-copula相依的两类风险模型

发布时间:2019-06-22 13:10
【摘要】:在随机过程及其应用的领域,关于保险风险模型的文章是数不胜数,人们也不再仅仅满足于对古典风险模型的研究.到目前为止,更新风险模型,带扰动的风险模型,索赔过程相依的模型,保费随机化的模型,对偶模型等都得到了广泛地研究.这篇论文讨论一种新型的相依模型,即在保费随机化模型的基础上对于保费收取过程做进步推广,我们假定保险公司收取的保费和收取保费的时间间隔是相依的,这种相依结构是通过FGM-copula函数建立的,索赔过程仍是复合泊松过程.本文主要考虑两类风险模型,第一类是在保费随机化模型基础上直接拓展而来的风险模型,第二类是对第一类模型的进一步推广,即带扰动的风险模型,对这两类模型的研究在保险实务中有着非常重要的意义. 本文共分为三部分. 第一部分主要介绍了文章所研究的两类风险模型以及有关的背景知识.第一类模型是:第二类模型是: 第二部分讨论了第一类风险模型的两类分红策略:障碍分红策略和阈值分红策略.在障碍分红策略下,得到期望折扣罚金函数mb,δ(u;b))满足的积分方程为:(λ1+λ2+δ)mb,δ(u;b)=λ2∫0u mb,δ(u-y;b)dF(y)+λ2∫u∞w(w,y-u)dF(y)+∫(b-u)0mb,δ(u+x;b)gx,w(x,0)dx+∫∞(b-u) mb,∧(b;b)gx,w(x,0)dx,分红函数V(u:b)所满足的积分方程为:(λ1+λ2+δ)V(u;b)=λ2∫u0V(u-y;b)dF(y)+∫(b-u)0V(u+x;b)gx,w(x,0)dx+∫∞b-u(u+x-b+V(b;b))gx,w(x,0)dx.在索赔额和保费收取额都服从指数分布这种特殊情形下,我们又对其进行了相关探讨,相继得到了破产时的Laplace变换和分红函数所满足的方程以及它们的具体表达式.在此基础上,我们可以求出保费收取额与保费收取时间间隔相互独立情形下的些相关量的具体表达式,得到的结果与之前的结论相吻合.在阈值分红策略下,分红函数满足的积分微分方程为:当0ub时,(λ1+λ2+δ)V1(u;b)=λ2∫u0V1(u-y;b)dF(y)+∫b-u0V1(u+x;b)gx,w(x,0)dx+∫∞b-n V2(u+x;b)gx,w(x,0)dx,当ub时,(λ1+λ2+δ)V2(u;b)=α-αV'2(u;b)+λ2∫u-00V2(u-y;b)dF(y) λ2∫αu-b V1(u-y;b)dF(y)+∫∞0V2(u+x;b)gx,w(x,0)dx.然后就索赔额和保费收取额都服从指数分布的特殊情形,本部分以推论的形式给出了结果. 第三部分主要讨论第二类风险模型.类似地,我们求出它在障碍分红策略下的分红函数满足的积分微分方程为:(λ1+λ2+δ)Vσ(u;b)=1/2σ2V"σ(u;b)+λ2∫u0Vσ(u-y;b)dF(y)+∫b-u0Vσ(u+x;b)gx,w(x,0)dx+∫∞b-u(u+x-b+Vσ(b;b))gx,w(x,0)dx.在阈值分红策略下的分红函数所满足的积分-微分方程为:当0ub时,(λ1+λ2+δ)V1,σ(u;b)=(σ2)/2V"1,σ(u;b)+λ2∫u0V1,σ(u-y;b)dF(y)+∫b-u0V1,σ(u+x;b)gx,w(x,0)dx∫∞b-uV2,σ(u+x;b)当ub时,(λ1+λ2+δ)V2,σ(u;b)=α-βV'2,σ(u;b)+(σ2)/2V"2,σ(u;b)+λ1∫u-b0V2,σ(u-y;b)dF(y)+λ2∫u u-b V1,σ(u-y;b)dF(y)+∫∞0V2,σ(u+x;b)gx,w(x,0)dx.除此之外,对于类似于第二部分的特殊情形,本文也进行了详细的讨论. 本文主要讨论了所研究模型在特殊情形下的分红函数的具体表达形式,而对于更一般的情形,还尚待解决.
[Abstract]:In the field of the stochastic process and its application, the article on the insurance risk model is numerous, and people are no longer satisfied with the study of the classical risk model. So far, the model of risk model, risk model with disturbance, model of claim process, model of premium randomization, dual model and so on have been widely studied. This paper discusses a new dependent model, that is, on the basis of the model of premium randomization, we assume that the insurance premium and the time interval of the premium are dependent on the premium collection process, which is established by the FGM-coula function. The process of claim is still a compound Poisson process. In this paper, two types of risk models are considered, the first is the risk model developed directly on the basis of the model of premium randomization, and the second is the further extension of the first model, i. e. the risk model with disturbance, The study of these two models is of great importance in the practice of insurance. This paper is divided into three parts: The first part mainly introduces two types of risk models and the related back of the article. View knowledge. The first model is: the second category The model is that the second part discusses two types of bonus strategies of the first type of risk model: the strategy and the threshold of the bonus. The value bonus strategy is as follows: (1 + (2 + 1) mb, (u; b) = {2} {0 u mb,} (u; b) = {2} {0 u mb,} (u-y; b) dF (y) + {2} u (w, y-u) dF (y) + {2} u (w, y-u) dF (y) + 2 (b-u) 0mb, under an obstacle bonus strategy. (u + x; b) gx, w (x,0) dx + __ (b-u) mb, w (b; b) gx, w (x,0) dx, the integral equation satisfied by the bonus function V (u: b) is: (u; b) = {2} u0V (u-y; b) dF (y) + 2 (b-u)0 V (u + x; b) gx, w (x,0) dx + {\ b-u (u + x-}) b+V(b;b))gx,w(x (0) dx. In this special case, the amount of the claim and the amount of the premium are subject to the exponential distribution. In this special case, we have also discussed it, and the equations of the Laplace transform and the bonus function at the time of the bankruptcy have been obtained. On this basis, we can find the specific expression of some correlation between the premium collection amount and the premium collection time interval, and the result is the same as that of the previous The integral differential equation that is satisfied by the bonus function under the threshold bonus strategy is: (1 + 2 + 1) V1 (u; b) = {2} u0V1 (u-y; b) dF (y) + b-u0V1 (u + x; b) dF (y) + b-u0V1 (u + x; b) gx, w (x,0) dx + {\ b-n V2 (u + x; b) gx, w (x,0) dx , when the hub is, (+ 1 + {2 +}) V2 (u; b) = 1-2 V '2 (u; b) + {2} u-00 V2 (u-y; b) dF (y) = 2} {u-b V1 (u-y; b) dF (y) + {= 0 V2 (u + x; b) gx, w (x ,0) dx. The amount of the claim and the amount of the premium are then subject to a special case of an exponential distribution, in the form of an inference. The results are given. The third part mainly discusses The second type of risk model. Similarly, we find the integral differential equation for which the bonus function under the obstacle bonus strategy is: (1 + 2 + 2 + 1) V (u; b) = 1/2,2 V "(u; b) + {2} u0V (u-y; b) dF (y) + {b-u0V} (u + x; b) gx, w (x,0) dx + {\ b-u (u + x-b + V} (b; b)) gx, w (x,0) dx. The integral-differential equation satisfied by the bonus function under the threshold bonus strategy is: when 0 ub, (1 + 2 + 1) V1,1 (u; b) = (Sup2)/2 V" 1,1 (u; b) + {2} u0V1,1 (u-y; b) dF (y) + {b-u0V1,} (u + x; b) ) gx, w (x,0) dx {\ b-uV2,} (u + x; b) when b, (u; b) = 1-{2 +} V2,} (u; b) = 1-{V '2,} (u; b) + ({2)/2 V "2,} (u; b) + {1} u-b0V2,} (u-y; b) dF (y) + {2} u-b V 1, u (u-y; b) dF (y) + {= 0 V2,} (u + x; b) gx, w (x,0) dx. In addition, this article also The detailed discussion is given. This paper mainly discusses the specific expression of the bonus function of the model under special circumstances, and for more general
【学位授予单位】:曲阜师范大学
【学位级别】:硕士
【学位授予年份】:2014
【分类号】:O211.67;F840.4

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1 姚定俊;汪荣明;徐林;;随机保费风险模型下的平均折现罚金函数(英文)[J];应用概率统计;2008年03期



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