带干扰风险模型的破产问题研究
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摘要
古典风险模是人们最早提出的风险模型,也是研究的最为透彻的模型.但该模型过于理想化,现实中有许多干扰项,因此研究带干扰的风险模型是有必要的. Gerber-Shiu罚金折现函数自从1998年由Gerber和Shiu引入到风险理论中以来,一直是人们研究风险理论的热点问题.本文研究了带干扰风险模型的Gerber-Shiu罚金函数和有关的联合分布等问题.
本文共分五章:
第一章介绍了风险模型发展的历史和现状.
第二章介绍了部分有关的概率论知识.
第三章针对Barrier分红策略下的干扰风险模型研究了Gerber-Shiu罚金折现函数,首先,利用布朗运动的有关知识首先获得了Gerber-Shiu罚金折现函数满足的积分方程
其中(?),
然后在此基础上利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.
第四章在第三章的基础上进一步研究了Barrier分红策略下的带有常数利率的干扰风险模型,首先运用布朗运动的知识我们得到了该模型的Gerber-Shiu罚金折现函数的积分方程
其中(?).
然后利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.
最后,我们应用It(?)公式求得了Gerber-Shiu罚金折现函数所满足的积分一微分方程
第五章研究了一类随机保费的干扰风险模型的破产问题.设保费率为一任意的离散随机变量ξ(ω),取值为k_i,i=1,2,…,p_i=P(ξ(ω)=k_i).则风险模型可表示为
其中ξ(ω)=k_i时,U_i(t):=u+k_it-(?)+W(t), t≥0为带干扰的经典风险模型.
我们用随机过程和概率论的方法推导破产概率、末离前最大盈余分布、破产时、破产前瞬时盈余与破产时赤字的联合分布等精算量分布的具体表达式.
The classical risk model,which was put forward earlier than other models has been studied most thorough. But it's an ideal model because there are many disturbances in real life. So it's very necessary to research the risk model perturbed by diffusion. Gerber-Shiu discounted penalty function became a hot topic from 1998,when it was introduced in risk theories. In this paper, Gerber-Shiu discounted penalty function and joint distributions of some actuarial random variables are discussed.
The article is divided into 5 sections.
Chapter 1 is about the development and current situation of risk model.
Chapter 2 introduces some related stochastic process theories.
Chapter 3 discusses Gerber-Shiu discounted penalty function in the risk model perturbed by diffusion with Barrier dividend strategy. First, the integral equation of Gerber-Shiu discounted penalty function is obtained from the knowledge of Brownian motion
where t_0≤(?),0≤a≤(?),
On this basis, the properties of continuity and twice-continuous differentiability for m_s(u) are also got from the boundness of m_s(u) and continuous differentiability of the functions H, h and F.
Chapter 4 is founded on chapter 3. In this part, we make a further research of the risk model perturbed by diffusion with constant interest rate under Barrier dividend strategy. First, we get the integral equation of Gerber-Shiu discounted penalty function from the knowledge of Brownian motion
Where
Then the twice-continuous differentiability of Gerber-Shiu discounted penalty functionis discussed by the boundness of m_s(u) and continuous differentiability of the functions H, h and F.
At last, we get its integro-differential equation by the use of Ito's formula
In chapter 5, ruin problem of risk model perturbed by diffusion with stochastic premium rate is considered. Suppose the premium rate is any discrete random variableξ(ω), whose values are k_i, i = 1,2,…. Then the expression of the risk model is
Whenξ(ω)=k_i,U_i(t)=u+k_it-(?)X_i+W(t) for t≥0 is the classical modelperturbed by diffusion.
Then the exact expressions for actuarial diagnostics, such as the ruin probability, the distribution of extreme surplus before ruin, the joint distribution of the surplus immediately before ruin, the deficit at ruin and the ruin time, are concluded through stochastic process and probability theory methods.
本文共分五章:
第一章介绍了风险模型发展的历史和现状.
第二章介绍了部分有关的概率论知识.
第三章针对Barrier分红策略下的干扰风险模型研究了Gerber-Shiu罚金折现函数,首先,利用布朗运动的有关知识首先获得了Gerber-Shiu罚金折现函数满足的积分方程
其中(?),
然后在此基础上利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.
第四章在第三章的基础上进一步研究了Barrier分红策略下的带有常数利率的干扰风险模型,首先运用布朗运动的知识我们得到了该模型的Gerber-Shiu罚金折现函数的积分方程
其中(?).
然后利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.
最后,我们应用It(?)公式求得了Gerber-Shiu罚金折现函数所满足的积分一微分方程
第五章研究了一类随机保费的干扰风险模型的破产问题.设保费率为一任意的离散随机变量ξ(ω),取值为k_i,i=1,2,…,p_i=P(ξ(ω)=k_i).则风险模型可表示为
其中ξ(ω)=k_i时,U_i(t):=u+k_it-(?)+W(t), t≥0为带干扰的经典风险模型.
我们用随机过程和概率论的方法推导破产概率、末离前最大盈余分布、破产时、破产前瞬时盈余与破产时赤字的联合分布等精算量分布的具体表达式.
The classical risk model,which was put forward earlier than other models has been studied most thorough. But it's an ideal model because there are many disturbances in real life. So it's very necessary to research the risk model perturbed by diffusion. Gerber-Shiu discounted penalty function became a hot topic from 1998,when it was introduced in risk theories. In this paper, Gerber-Shiu discounted penalty function and joint distributions of some actuarial random variables are discussed.
The article is divided into 5 sections.
Chapter 1 is about the development and current situation of risk model.
Chapter 2 introduces some related stochastic process theories.
Chapter 3 discusses Gerber-Shiu discounted penalty function in the risk model perturbed by diffusion with Barrier dividend strategy. First, the integral equation of Gerber-Shiu discounted penalty function is obtained from the knowledge of Brownian motion
where t_0≤(?),0≤a≤(?),
On this basis, the properties of continuity and twice-continuous differentiability for m_s(u) are also got from the boundness of m_s(u) and continuous differentiability of the functions H, h and F.
Chapter 4 is founded on chapter 3. In this part, we make a further research of the risk model perturbed by diffusion with constant interest rate under Barrier dividend strategy. First, we get the integral equation of Gerber-Shiu discounted penalty function from the knowledge of Brownian motion
Where
Then the twice-continuous differentiability of Gerber-Shiu discounted penalty functionis discussed by the boundness of m_s(u) and continuous differentiability of the functions H, h and F.
At last, we get its integro-differential equation by the use of Ito's formula
In chapter 5, ruin problem of risk model perturbed by diffusion with stochastic premium rate is considered. Suppose the premium rate is any discrete random variableξ(ω), whose values are k_i, i = 1,2,…. Then the expression of the risk model is
Whenξ(ω)=k_i,U_i(t)=u+k_it-(?)X_i+W(t) for t≥0 is the classical modelperturbed by diffusion.
Then the exact expressions for actuarial diagnostics, such as the ruin probability, the distribution of extreme surplus before ruin, the joint distribution of the surplus immediately before ruin, the deficit at ruin and the ruin time, are concluded through stochastic process and probability theory methods.
引文
[1]Filip Lundberg.Approximerad framstallning av sannolikhetsfunktionen[J].1903.
[2]Dufrcsne,F.,Gerber,H.U.Risk theory for the compound Poisson process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,1991,10:51-59.
[3]Chiu,S.N.,Yin,C.C,The time of ruin,the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion[J].Insurance:Mathematics and Economics,2003,33:59-66.
[4]Chunsheng Zhang,Guojing Wang.The joint density function of three characteristics on jump-diffusion risk process[J],Insurance:Mathematics and Economics,2003,32:445-455.
[5]Gerber,H.U.,Landry,B.On the discounted penalty at ruin in a jump-diffusion and the perpetual put option[J].Insurance:Mathematics and Economics,1998,22:263-276.
[6]Guojing Wang,Rong Wu.Some distributions for classical risk process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,2000,26:15-24.
[7]Tsai,C.C.L.,Willmot,G.E.,A generalized defective renewal equation for the surplus process perturbed by diffusion[J].Insurance:Mathematics and Economics,2002a,30:51-66.
[8]Tsai,C.C.L.,On the expectations of the present values of the time of ruin perturbed by diffusion[J].Insurance:Mathematics and Economics,2003,32:413-429.
[9]Jun Cai.Ruin probabilities and penalty functions with stochastic rates of interest [J].Stochastic Processes and their Applications,2004,112:53-78.
[10]Guojing Wang,Rong Wu.Distributions for the risk process with a stochastic return on investments [J].Stochastic Processes and their Applications,2001,95:329-341.
[11]Paulsen,J.Risk theory in a stochastic economic environmentfj].Stochastic Processes and their Applications,1993,46:327-361.
[12]Paulsen,J.,Gjessing,H.K.Ruin theory with stochastic return on investments [J].Advances in Applied Probability,1997,29:965-985.
[13]Paulsen,J.,Kasozi,J.and Steigen,A.A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments [J].Insurance:Mathematics and Economics,2005,36:399-420.
[14]Dickson,David CM.On a class of renewal risk process[J].North American Actuarial Journal.1998,2(3):60-68;Discussions:68-73.
[15]Dickson,David CM.and Christian Hipp.On the time to ruin for Erlang(2)risk processes[J].Insurance:Mathematics and Economics,2001,29:333-344.
[16]Gerber,H.U.,Shiu,E.S.W.The time value of ruin in a Sparre Andersen model[J].North American Actuarial Journal,2005,9:49-69.
[17]David Landriault.,Gordon Willmot.On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution [J].Insurance Mathematics and Economics,2008,42:600-608.
[18]Durrett,R.Brownian motion and martingales in analysis[M].Wadsworth,Belmont,California.1984.
[19]Gerber,H.U.An extension of the renewal equation and its application in the collective theory of risk[J].Scandinavian Actuarial Journal,1970:205-210.
[20]S Li,J Garrido.On a general class of renewal risk process:analysis of the Gerber-Shiu function[J].Advances in Applied Probability,2005,37:836-856.
[21]Kam C.Yuen,Guojing Wangb and Wai K.Li.The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barricr[J].Insurance:Mathematics and Economics.2007.40:104-112.
[22]Gerber H.U.and Shiu E.S.W.Optimal dividends:Analysis with Brownian motion[J].North American Actuarial Journal.2004,8(1):l-20.
[23]Gerber H.U.and Shiu E.S.W.On optimal dividends:From reflection to refraction[J].Journal of Computational and Applied Mathematics.2006,186:4-22.
[24]Dickson D.C.M.and Waters H.R.Some optimal dividends problems[J].Astin Bulletin.2004,34:49-74.
[25]Karaztas,I.,Shreve,S.E.Brownian motion and stochastic calculus[M].Springer-Verlag,New York.1991.
[26]Li,Shuanming.The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion [J].Scandinavian Actuarial Journal,2006(2):73-85.
[27] Port, S., Stone, C. Brownian motion and classical potential theory[M]. New York: Academic press. 1978.
[28] Revuz, D., Yor, M. Continuous martingales and Brownian motion[M]. Berlin: Springer-Verlag. 1991.
[29] 刘家军,刘再明.保费到达为更新过程的复合更新风险模型[J].数学理论与应用.2003年第01期:102-106.
[30] 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[J].数学物理学报(A辑).2007,27(4).
[31] 毕秀春,王新华,李荣.带干扰古典风险模型的极值联合分布[J].曲阜师范大学学报,2004,30(2):19-23.
[32] 蔡高玉,耿显民.一类随机保费率下的风险模型[J].应用数学与计算数学学报,2007,21(1):27-33.
[33] 吴荣,王过京.带干扰经典风险过程在破产时刻的余额分布[J].南开大学学报(自然科学),1999,32(3):25-29。
[2]Dufrcsne,F.,Gerber,H.U.Risk theory for the compound Poisson process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,1991,10:51-59.
[3]Chiu,S.N.,Yin,C.C,The time of ruin,the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion[J].Insurance:Mathematics and Economics,2003,33:59-66.
[4]Chunsheng Zhang,Guojing Wang.The joint density function of three characteristics on jump-diffusion risk process[J],Insurance:Mathematics and Economics,2003,32:445-455.
[5]Gerber,H.U.,Landry,B.On the discounted penalty at ruin in a jump-diffusion and the perpetual put option[J].Insurance:Mathematics and Economics,1998,22:263-276.
[6]Guojing Wang,Rong Wu.Some distributions for classical risk process that is perturbed by diffusion[J].Insurance:Mathematics and Economics,2000,26:15-24.
[7]Tsai,C.C.L.,Willmot,G.E.,A generalized defective renewal equation for the surplus process perturbed by diffusion[J].Insurance:Mathematics and Economics,2002a,30:51-66.
[8]Tsai,C.C.L.,On the expectations of the present values of the time of ruin perturbed by diffusion[J].Insurance:Mathematics and Economics,2003,32:413-429.
[9]Jun Cai.Ruin probabilities and penalty functions with stochastic rates of interest [J].Stochastic Processes and their Applications,2004,112:53-78.
[10]Guojing Wang,Rong Wu.Distributions for the risk process with a stochastic return on investments [J].Stochastic Processes and their Applications,2001,95:329-341.
[11]Paulsen,J.Risk theory in a stochastic economic environmentfj].Stochastic Processes and their Applications,1993,46:327-361.
[12]Paulsen,J.,Gjessing,H.K.Ruin theory with stochastic return on investments [J].Advances in Applied Probability,1997,29:965-985.
[13]Paulsen,J.,Kasozi,J.and Steigen,A.A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments [J].Insurance:Mathematics and Economics,2005,36:399-420.
[14]Dickson,David CM.On a class of renewal risk process[J].North American Actuarial Journal.1998,2(3):60-68;Discussions:68-73.
[15]Dickson,David CM.and Christian Hipp.On the time to ruin for Erlang(2)risk processes[J].Insurance:Mathematics and Economics,2001,29:333-344.
[16]Gerber,H.U.,Shiu,E.S.W.The time value of ruin in a Sparre Andersen model[J].North American Actuarial Journal,2005,9:49-69.
[17]David Landriault.,Gordon Willmot.On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution [J].Insurance Mathematics and Economics,2008,42:600-608.
[18]Durrett,R.Brownian motion and martingales in analysis[M].Wadsworth,Belmont,California.1984.
[19]Gerber,H.U.An extension of the renewal equation and its application in the collective theory of risk[J].Scandinavian Actuarial Journal,1970:205-210.
[20]S Li,J Garrido.On a general class of renewal risk process:analysis of the Gerber-Shiu function[J].Advances in Applied Probability,2005,37:836-856.
[21]Kam C.Yuen,Guojing Wangb and Wai K.Li.The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barricr[J].Insurance:Mathematics and Economics.2007.40:104-112.
[22]Gerber H.U.and Shiu E.S.W.Optimal dividends:Analysis with Brownian motion[J].North American Actuarial Journal.2004,8(1):l-20.
[23]Gerber H.U.and Shiu E.S.W.On optimal dividends:From reflection to refraction[J].Journal of Computational and Applied Mathematics.2006,186:4-22.
[24]Dickson D.C.M.and Waters H.R.Some optimal dividends problems[J].Astin Bulletin.2004,34:49-74.
[25]Karaztas,I.,Shreve,S.E.Brownian motion and stochastic calculus[M].Springer-Verlag,New York.1991.
[26]Li,Shuanming.The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion [J].Scandinavian Actuarial Journal,2006(2):73-85.
[27] Port, S., Stone, C. Brownian motion and classical potential theory[M]. New York: Academic press. 1978.
[28] Revuz, D., Yor, M. Continuous martingales and Brownian motion[M]. Berlin: Springer-Verlag. 1991.
[29] 刘家军,刘再明.保费到达为更新过程的复合更新风险模型[J].数学理论与应用.2003年第01期:102-106.
[30] 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[J].数学物理学报(A辑).2007,27(4).
[31] 毕秀春,王新华,李荣.带干扰古典风险模型的极值联合分布[J].曲阜师范大学学报,2004,30(2):19-23.
[32] 蔡高玉,耿显民.一类随机保费率下的风险模型[J].应用数学与计算数学学报,2007,21(1):27-33.
[33] 吴荣,王过京.带干扰经典风险过程在破产时刻的余额分布[J].南开大学学报(自然科学),1999,32(3):25-29。