非时齐复合Poisson风险模型的破产特征量分析

摘要/Abstract

摘要：

该文将经典风险模型推广到非时齐复合Poisson风险模型.首先，运用经典方法和时变方法，计算了该模型下的破产特征量，且得到了更新方程的解析表达式.其次，定义了时变后相应模型的一个广义的Gerber-Shiu函数，验证了时变方法对非时齐Poisson风险模型的有效性.最后，当单次索赔量服从指数分布时，计算了相应的破产概率和Gerber-Shiu函数.

关键词: 破产概率, 时变方法, 非时齐Poisson过程, Gerber-Shiu函数, 更新方程

Abstract:

In this paper, the classical risk model is extended to nonhomogeneous compound Poisson risk model. Firstly, both the classical method and the time-varying method are used to calculate the ruin-related quantities for this model, and the analytical expression of the renewal equation is obtained. Secondly, for the time-varying model, the generalized Gerber-Shiu function is defined, which is to verify the effectiveness of the time-varying method for the nonhomogeneous compound Poisson risk model. Finally, when each claim follows an exponentially distribution, the corresponding ruin probability and Gerber-Shiu function are calculated.

Key words: Ruin probability, Time-varying method, Nonhomogeneous Poisson process, Gerber-Shiu function, Renewal equation

中图分类号:

O211.6

邓迎春,李满,黄娅,周杰明. 非时齐复合Poisson风险模型的破产特征量分析[J]. 数学物理学报, 2020, 40(2): 501-514.

Yingchun Deng,Man Li,Ya Huang,Jieming Zhou. On the Analysis of Ruin-Related Quantities in the Nonhomogeneous Compound Poisson Risk Model[J]. Acta mathematica scientia,Series A, 2020, 40(2): 501-514.

参考文献 40

1	Lundberg F . On the theory of reinsurance. Transactions Ⅵ International Congress of Actuaries, 1909, 1, 877- 948
2	Cramér H. On the Mathematical Theory of Risk. Stockholm: Skandia Jubilee Volume, 1930
3	Cramér H. Collective Risk Theory: Asurvey of the Theory from the Point of View of the Theory of Stochastic Processes. Stockholm: Nordiska Bokhandeln, 1955
4	Yuen K C , Guo J Y . Ruin probabilities for time-correlated claims in the compound binomial model. Insurance Mathematics and Economics, 2001, 29 (1): 47- 57 doi: 10.1016/S0167-6687(01)00071-3
5	Gerber U , Yang H L . Absolute ruin probabilities in a jump diffusion model with investment. North American Actuarial Journal, 2007, 11 (3): 159- 169 doi: 10.1080/10920277.2007.10597474
6	Li J C , Dickson D C M , Li S M . Some ruin problems for the MAP risk model. Insurance Mathematics and Economics, 2015, 65, 1- 8 doi: 10.1016/j.insmatheco.2015.08.001
7	Li J C , Dickson D C M , Li S M . Analysis of some ruin-related quantities in a Markov-modulated risk model. Stochastic Models, 2016, 32 (3): 351- 365 doi: 10.1080/15326349.2015.1121399
8	Liu R F , Wang D , Peng J . Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial and Management Optimization, 2017, 13 (2): 995- 1007 doi: 10.3934/jimo.2016058
9	Li J Z . The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims. Communications in Statistics-Theory and Methods, 2017, 46 (4): 1959- 1971 doi: 10.1080/03610926.2015.1030428
10	尹传存. 关于破产概率的一个局部定理. 中国科学(A辑), 2004, 34 (2): 192- 202
	Yin C C . A local theorem about the ruin probability. Science in China Series A, 2004, 34 (2): 192- 202
11	刘艳, 胡亦钧. 马氏环境下带扰动的Cox相关的风险模型破产概率的上界估计. 数学年刊, 2004, 25 (5): 579- 586
	Liu Y , Hu Y J . Upper bound estimation on the ruin probability for a cox correlated risk model disturbed by diffusion in a markovian environment. Chinese Journal of Contemporary Mathematies, 2004, 25 (5): 579- 586
12	Konstantinides D , Tang Q H , Tsitsiashvili G . Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails. Insurance Mathematics and Economics, 2002, 31 (3): 447- 460 doi: 10.1016/S0167-6687(02)00189-0
13	唐启鹤. 重尾索赔下关于破产概率的一个等价式. 中国科学(A辑), 2002, 32 (3): 260- 266
	Tang Q H . An equivalent formula of the ruin probability under a heavy tail claim. Science China Series A, 2002, 32 (3): 260- 266
14	郭风龙, 王定成, 张维. 考虑随机投资收益和单边线性索赔的时间依赖更新风险模型的破产概率. 中国科学(A辑), 2016, 46A (3): 321- 350
	Guo F L , Wang D C , Zhang W . Ruin probability of renewal risk model with stochastic investment returns and one-sided linear time-dependent claims. Science China Series A, 2016, 46A (3): 321- 350
15	Wang L . Asymptotic behavior of finite-time ruin probability in a by-claim risk model with constant interest rate. Journal of Mathematics and Statistics, 2014, 10 (3): 339- 357
16	Xie J H , Zou W . Dividend barrier and ruin problems for a risk model with delayed claims. Communications in Statistics-Theory and Methods, 2017, 46 (14): 7063- 7084 doi: 10.1080/03610926.2016.1143010
17	Constantinescu C , Samorodnitsky G , Zhu W . Ruin probabilities in classical risk models with gamma claims. Scandinavian Actuarial Journal, 2018, 2018 (7): 555- 575 doi: 10.1080/03461238.2017.1402817
18	Oshime T , Shimizu Y . Parametric inference for ruin probability in the classical risk model. Statistics and Probability Letters, 2018, 133, 28- 37 doi: 10.1016/j.spl.2017.09.020
19	Gerber U , Shiu S . On the time value of ruin. North American Actuarial Journal, 1998, 2 (1): 48- 72 doi: 10.1080/10920277.1998.10595671
20	Lin X S , Willmot G E . Analysis of a defective renewal equation arising in ruin theory. Insurance Mathematics and Economics, 1999, 25 (1): 63- 84 doi: 10.1016/S0167-6687(99)00026-8
21	Lin X S , Willmot G E , Drekic S . The classical risk model with a constant dividend barrier:analysis of the Gerber-Shiu discounted penalty function. Insurance Mathematics and Economics, 2004, 33 (3): 551- 566
22	Albrecher H , Boxma O J . A ruin model with dependence between claim sizes and claim intervals. Insurance Mathematics and Economics, 2004, 35 (2): 245- 254 doi: 10.1016/j.insmatheco.2003.09.009
23	Albrecher H , Boxma O J . On the discounted penalty function in a Markov-dependent risk model. Insurance Mathematics and Economics, 2005, 37 (3): 650- 672 doi: 10.1016/j.insmatheco.2005.06.007
24	Albrecher H , Cheung E C K , Thonhauser S . Randomized observation periods for the compound Poisson risk model:the discounted penalty function. Scandinavian Actuarial Journal, 2013, 2013 (6): 424- 452 doi: 10.1080/03461238.2011.624686
25	Zhang Z M , Su W . A new efficient method for estimating the Gerber-Shiu function in the classical risk model. Scandinavian Actuarial Journal, 2018, 2018 (5): 426- 449 doi: 10.1080/03461238.2017.1371068
26	Willmot G E, Woo J K. Gerber-Shiu analysis in the classical Poisson risk model//Willmot G E, Woo J K. Surplus Analysis of Sparre Andersen Insurance Risk Processes. New York: Springer, 2017: 45-59
27	Zhang Z M , Yang Y , Liu C L . On a perturbed compound Poisson model with varying premium rates. Journal of Industrial and Management Optimization, 2017, 13 (2): 721- 736 doi: 10.3934/jimo.2016043
28	Li S L , Yin C C , Zhao X , Dai H S . Stochastic interest model based on compound Poisson process and applications in actuarial science. Mathematical Problems in Engineering, 2017, 2, 1- 8
29	韦晓, 于金酉, 胡亦钧. 变保费率扰动风险模型的有限时间破产概率和大偏差. 数学物理学报, 2007, 27A (4): 616- 623
	Wei X , Yu J Y , Hu Y J . Large deviations and finite time ruin probability for perturbed risk model with variable premium rate(in Chinese). Acta Mathematica Scientia, 2007, 27A (4): 616- 623
30	Picard P . A nonhomogeneous risk model for insurance. Computers and Mathematics with Applications, 2006, 51 (2): 325- 334 doi: 10.1016/j.camwa.2005.11.005
31	Kartashov M . Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates. Theory of Probability and Mathematical Statistics, 2009, 78, 61- 73 doi: 10.1090/S0094-9000-09-00762-5
32	Kartashov M . Boundedness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis. Theory of Probability and Mathematical Statistics, 2010, 81, 71- 83 doi: 10.1090/S0094-9000-2010-00811-8
33	Blaževičius K , Bieliauskieně E , Šiaulys J . Finite-time ruin probability in the inhomogeneous claim case. Lithuanian Mathematical Journal, 2010, 50 (3): 260- 270 doi: 10.1007/s10986-010-9084-2
34	Bieliauskieně E , Šiaulys J . Infinite time ruin probability in inhomogeneous claims case. Lietuvos Matematikos Rinkinys, 2010, 51, 352- 356
35	Andrulyt I M , Bernackait E , Kievinait D , Šiaulys J . A Lundberg-type inequality for an inhomogeneous renewal risk model. Modern Stochastics:Theory and Applications, 2015, 2 (2): 173- 184
36	Abaurrea J , Asín J , Cebrián A C . Modeling and projecting the occurrence of bivariate extreme heat events using a non-homogeneous common Poisson shock process. Stochastic Environmental Research and Risk Assessment, 2015, 29 (1): 309- 322
37	Bernackaitě E , Šiaulys J . The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial and Management Optimization, 2017, 13 (1): 207- 222 doi: 10.3934/jimo.2016012
38	张波, 张景肖. 应用随机过程. 北京: 清华大学出版社, 2004: 42- 45
	Zhang B , Zhang J X . Applying Stochastic Processes. Beijing: Tsinghua University Press, 2004: 42- 45
39	杨琴, 陈云. 基于泊松过程的供应链复杂网络模型. 系统工程, 2012, 30 (9): 57- 62
	Yang Q , Chen Y . Supply chain complex network model based on poisson process. Systems Engineering, 2012, 30 (9): 57- 62
40	Albrecher H , Boxma O O , Essifi R R , Kuijstermans R . A queuing model with a randomized depletion of inventory. Probability in the Engineering and Informational Sciences, 2017, 31 (1): 43- 59 doi: 10.1017/S0269964816000322

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