风险模型的最优投资和再保险
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摘要
在保险实务中,保险公司面临着诸多现实存在和潜在的各种风险。尽管无法预测或完全防范风险的发生,但可以通过购买再保险来转移和分散风险。同时,也可以通过在金融市场上投资来增加自己的财富,但投资是有风险的,再保险也需要分一部分保费给再保险公司。因此,研究最优投资和最优再保险问题,对保险公司来说,具有很重要的现实意义。
本论文主要研究随机控制在风险理论中的应用,通过建立动态规划模型来解决最优投资和再保险问题。首先对扩散风险模型中的红利分配引入投资和再保险,在边界分红方式下,得到使得期望红利最大的最优投资和比例再保险策略,以及最大期望红利的显示表达式；并通过数值计算给出了投资和再保险对期望红利的影响。其次,对跳-扩散风险模型,在盈余投资于风险资产和无风险资产,以及索赔进行比例再保险的策略下,分别在含交易费用和风险资产价格服从几何Levy过程下,研究使得保险公司期望效用最大的最优投资和比例再保险策略。得到最优投资和比例再保险策略,以及最大期望效用函数的显示表达式。最后通过数值分析找到最优投资和再保险策略与各参数之间的关系。
In insurance practice,the insurance company faces many reality risks and potential risks.Although impossible to predict or completely prevent the occurrence of risks,but they can divert and diverge their risks by purchasing reinsurance.At the same time,the insurance company can also be invested in the financial markets to increase wealth.Apparently, risky investment can be dangerous and reinsurance also need to diverge part of the premiums to the reinsurance company.Therefore,it is very important practical significance for the insurance company to study the problem of optimal investment and optimal reinsurance policy.
In this paper, we study the application of stochastic optimal control theory on the risk theory. We solve the problems of optimal investment and reinsurance policies by the establishment of dynamic programming model.Firstly,we introduce investment and reinsurance for diffusion risk model with dividends.Under barrier dividend strategy, the close form expressions for maximal expected discount dividend and the optimal investment and reinsurance policies are obtained. The impact of investment and reinsurance on dividend are also given by numerical calculation.Secondly, the surplus process is assumed to follow a jump-diffusion risk process.The insurance company has the possibility to invest in a risk-free asset and a risky asset and to purchase porportional reinsurance for claims.We study the optimal investment and porportional reinsurance policies which maximizing the expected utility of insurance company under the risky asset following Geometric Levy process and transaction cost, respectively. Explicit expressions for the maximal expected exponential utility and the corresponding optimal policies are obtained.Finally, the relationships between optimal investment and reinsurance policies and some parameters are given by numerical calculation.
本论文主要研究随机控制在风险理论中的应用,通过建立动态规划模型来解决最优投资和再保险问题。首先对扩散风险模型中的红利分配引入投资和再保险,在边界分红方式下,得到使得期望红利最大的最优投资和比例再保险策略,以及最大期望红利的显示表达式；并通过数值计算给出了投资和再保险对期望红利的影响。其次,对跳-扩散风险模型,在盈余投资于风险资产和无风险资产,以及索赔进行比例再保险的策略下,分别在含交易费用和风险资产价格服从几何Levy过程下,研究使得保险公司期望效用最大的最优投资和比例再保险策略。得到最优投资和比例再保险策略,以及最大期望效用函数的显示表达式。最后通过数值分析找到最优投资和再保险策略与各参数之间的关系。
In insurance practice,the insurance company faces many reality risks and potential risks.Although impossible to predict or completely prevent the occurrence of risks,but they can divert and diverge their risks by purchasing reinsurance.At the same time,the insurance company can also be invested in the financial markets to increase wealth.Apparently, risky investment can be dangerous and reinsurance also need to diverge part of the premiums to the reinsurance company.Therefore,it is very important practical significance for the insurance company to study the problem of optimal investment and optimal reinsurance policy.
In this paper, we study the application of stochastic optimal control theory on the risk theory. We solve the problems of optimal investment and reinsurance policies by the establishment of dynamic programming model.Firstly,we introduce investment and reinsurance for diffusion risk model with dividends.Under barrier dividend strategy, the close form expressions for maximal expected discount dividend and the optimal investment and reinsurance policies are obtained. The impact of investment and reinsurance on dividend are also given by numerical calculation.Secondly, the surplus process is assumed to follow a jump-diffusion risk process.The insurance company has the possibility to invest in a risk-free asset and a risky asset and to purchase porportional reinsurance for claims.We study the optimal investment and porportional reinsurance policies which maximizing the expected utility of insurance company under the risky asset following Geometric Levy process and transaction cost, respectively. Explicit expressions for the maximal expected exponential utility and the corresponding optimal policies are obtained.Finally, the relationships between optimal investment and reinsurance policies and some parameters are given by numerical calculation.
引文
[1]Asmussen S,Taksar M.Controlled diffusion models for optimal dividend payout [J].Insurance:Mathematics and Economics,1997,20:1-15.
[2]Buhlmann H.Mathematical Methods in Risk Theory [M].Springer, Berlin,1970.
[3]Browne S.Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin [J].Mathematics of Operations Research,1995,20:937-958.
[4]Cox J, Huang C.F.Optimal consumption and portfolio policies when asset prices follow a diffusion process [J].J.Econ.Theory,1989,49:33-83.
[5]Cadenillasa, Pliska S.Optimal trading of a security when there are taxes and transaction costs[J].Finance Stochastics,1999,3:137-165.
[6]Cai J,Dickson C.M.D.On the expected discounted penalty function at ruin of a surplus process with interest [J].Insurance:Mathematics and Economics,2002, 30:389-404.
[7]Cai J, Dickson C.M.D.Ruin probability with a markov chain interest model [J]. Insurance:Mathematics and Economics,2004,35:513-525.
[8]Cai J,Xu C.M.On the decomposition of the ruin probability for a jump-Diffusion surplus process compounded by a geometeic Brownian motion [J].North American Actuarial Journal,2006,10(20):120-132.
[9]Chiu S.N,Yin C.C.The time of ruin,the surplus prior to ruin and the deficite at ruin for the classical risk process perturbed by diffusion [J].Insurance: Mathematics and Economics,2003,33:59-66.
[10]De Finetti.Su un'impostazione alternativa dell teoria colletiva del rischio [J]. Transactions of the XV International Congress of Actuaries,1957,2:433-443.
[11]Dayananda P.W.A.Optimal reinsurance [J].Journal of Applied Probability, 1970,7:134-156.
[12]Dufersen F,Gerber H.U.Risk theory for the compound Poisson risk model process that is perturbed by diffusion [J].Insurance:Mathematical and Economics,1991,10:51-59.
[13]Dickson C.M.D.On the distribution of the surplus prior to ruin [J].Insurance: Mathematical and Economlcs,1992,7:191-207.
[14]Dickson D.C.M,Waters H.R. Some optimal dividend problem [J].ASTIN Bulletin,2004,4:49-74.
[15]Fleming W.H,Soner H.M.Controlled markove processes and viscosity solution [M].New York:Springer Verlag,1993.
[16]Gerber H.U.Entscheidigungskriterien fur den zusammengesetzten Poisson Prozess[J].Mitteilungen der Vereinigung Schweizer Versicherungs mathematiker 1969,19:185-228.
[17]Gerber H.U.An Extension to the Renewal Equation and its Application in the Collective Theory of Risk [J].Scandinavisk Aktuarietidskrift,1970,205-210.
[18]Gerber H.U.An Introduction to Mathematical Risk Theory [M].Philadephia,S.S. 1979.
[19]Gerber H.U,Shiu E.S.W..Optimal dividends:Analysis with Brownian motion [J].North American Actuarial Journal,2004,8 (1):1-20.
[20]Grandell J.Aspects of risk theory [M].New York:Springer-Verlag,1991.
[21]Grossman S.J,Zhou Z. Optimal investment strategies for controlling drawdowns [J].Math.Finance,1993,3:241-276.
[22]Guo J,Bai L.Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint [J].Insurance Mathematical and Economlcs,2008,42:968-975.
[23]Gao J.Optimal portfolios for DC pension plans under a CEV model [J]. Insurance:Mathematical and Economlcs,2009,44:479-490
[24]Hφjgaard B,Taksar M.Controlling risk exposure and dividend pay-out scemes: Insurance company example [J].Mathematical Finance,1999,9:153-182.
[25]Hojgaard B,Taksar M.Optimal risk control for a large corporation in the presence of returns on investments [J].Finance and Stochastics, 2001,5:527-547.
[26]Hipp C,Plum M.Optimal investment for insurer [J].Insurance:Mathematical and Economlcs 2000,27:215-228.
[27]Hipp C.Optimal investment for insurer:the Coxian case [J].Prerint,2002.
[28]Hipp C,Vogt M.Optimal dynamic XL reinsurance [J].Astin Bulletin, 2003,33(2):193-207.
[29]Hu Y,Zhinming Z.The perturbed compound Poisson risk model with multilayer dividend strategy [J].Statistic and Probability Letters,2008.
[30]Hu Y,ZhinmingZ.Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy [J].Insurance:Mathematic and Economics,2008,42:984-991.
[31]Irgens C, Paulsen J. Optimal control of risk exposure, reinsurance and investments for insurance portfolios [J].Insurance:Mathematics and Economics, 2004,35:21-51.
[32]Jeanblanc-Picque, Shiryaev A.N.Optimization of the flow of dividends [J].Russian Mathematical Surveys,1999,50:257-277.
[33]Lundberg F. I. Aterforsakring av kollektivrisker. Uppsala:Almqvist and Wiksell, 1903.
[34]Li J,Wu R.Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility [J].Applied Stochastic Model in Bussiness And Industry. Published Online:6 Feb 2009.
[35]Lin X.S,Willmot G.E, Drekic S.The classical risk model with a constant dividend barrier[J].Insurance:Mathematics and Economics,2003,33:551-566.
[36]Lin X.S,Pavlova K.P. The compound Poisson risk model with a thresholds strategy [J].Insurance:Mathematic and Economics,2006,38:57-80.
[37]Lin X.S,Pavlova K.P.The compound Poisson risk model with multiple thresholds [J].Insurance:Mathematic and Economics,2008,42:617-627.
[38]Li S.The distribution of the dividend payment in the compound Poisson risk model perturbed by diffusion. Working paper,2005.
[39]Maritina T.C,Gilber P.Stochastic control theory for optimal optimal investment.School of Actuarial Studies [J].Faculty of Commerce and Economics, 2003,1-23.
[40]Paulsen J.Optimal dividend payouts for diffusions with solvency constraints [J]. Finance and Stochastics,2003,4:457-474.
[41]Paulsen J,Gjessing H.K.Optimal choice of dividend barriers for a risk process with stochastic return on investments [J].Insurance:Mathematics and Economics 1997,20:215-223.
[42]Schmidli H.On the distribution of the surplus prior to and at ruin [J].Astin Bulletin,1999,29:227-244.
[43]Schmidli H.Optimal proportional reinsurance policy in a dynamic setting [J]. Scand Actuarial J,2001,55-68.
[44]Schmidli H.On minimizing the ruin probability by investment and insurance. Technical Report,No.175,Laboratory for Actuarial Mathematical,University of Copenhagen,2002,1-24.
[45]Schmidli H.Stochastic control in insyrance [M].Springer,London,2008.
[46]Shreve S,Soner HM.Optimal investment and consumption with transaction costs [J].Annals of Applied Probability,1994,4:609-692.
[47]Shreve S,Soner HM,Xu GL.Optimal investment and consumption with two bonds and transaction costs [J].Mathematical Finance,1991,1:53-84.
[48]Sundt B,Teugels J.L.The adjustment function in ruin estimates under interest force [J].Insurance:Mathematics and Economics,1997,19:85-94.
[49]Taksar M.Optimal risk and dividend distribution control model for an insurance company [J].Math Meth Operres,2000,51:1-42.
[50]Taksar M,Hogaard.B,Asmussen,S.Optimal risk Control and Dividend Distributlon Policies.Example of Exccss-of Loss Reinsurance [J].Finance and Stochastic,2000,4:299-324.
[51]Taksar M,Asmussen S.Controlled Diffusion models for Optimal Dividend pay-out [J].Insurance:Mathematics and Economics,1997,20:1-15.
[52]Wan Ning.Dividend payment with a threshold strategy in the compound Poisson risk perturbed by diffusion [J].Insurance:Mathematics and Economics,2007,40: 509-634.
[53]Xu G.L,Shreve S.E.A duality methods for optimal consumption and investment under short-selling prohibition:Ⅱ constant market coefficients [J].The Annals of Applied Probability,1992,2:314-328.
[54]Yang H, Zhang L.Optimal investment for insurer with jump-diffusion risk process [J].Insurance:Mathematics and Economics,2005,35:21-51.
[55]Zhang Xinli,Zhang Kecun,Yu xingjiang. Optimal proportional reinsurance and investment with transaction cost maximizing the terminal wealth [J].Insurance: Mathematical and Economics,2008,44:473-478.
[56]李秀芳,曾庆五.保险精算[M].中国会融出版社,200.5.
[57]梁志彬.跳扩散模型的最优投资和再保险[J].数学学报,2008,51：1195-1204.
[58]聂高秦.金融保险中的几类风险模型[M].华中科技大学,博士学位论文,2008
[59]粟芳.非寿险精算[M].清华大学出版社,2006.
[60]王爱香.一类超额损失和比例再保险的随机控制问题[M].上海大学,硕士学位论文,2006.
[61]徐林.具有随机投资收益的风险模型下若干破产问题的研究[M].华东师范大学,博士学位论文,2008.
[62]杨步清,叶中行.保险公司的最优再保险和红利分配[J].系统工程2000,6：23-27.
[2]Buhlmann H.Mathematical Methods in Risk Theory [M].Springer, Berlin,1970.
[3]Browne S.Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin [J].Mathematics of Operations Research,1995,20:937-958.
[4]Cox J, Huang C.F.Optimal consumption and portfolio policies when asset prices follow a diffusion process [J].J.Econ.Theory,1989,49:33-83.
[5]Cadenillasa, Pliska S.Optimal trading of a security when there are taxes and transaction costs[J].Finance Stochastics,1999,3:137-165.
[6]Cai J,Dickson C.M.D.On the expected discounted penalty function at ruin of a surplus process with interest [J].Insurance:Mathematics and Economics,2002, 30:389-404.
[7]Cai J, Dickson C.M.D.Ruin probability with a markov chain interest model [J]. Insurance:Mathematics and Economics,2004,35:513-525.
[8]Cai J,Xu C.M.On the decomposition of the ruin probability for a jump-Diffusion surplus process compounded by a geometeic Brownian motion [J].North American Actuarial Journal,2006,10(20):120-132.
[9]Chiu S.N,Yin C.C.The time of ruin,the surplus prior to ruin and the deficite at ruin for the classical risk process perturbed by diffusion [J].Insurance: Mathematics and Economics,2003,33:59-66.
[10]De Finetti.Su un'impostazione alternativa dell teoria colletiva del rischio [J]. Transactions of the XV International Congress of Actuaries,1957,2:433-443.
[11]Dayananda P.W.A.Optimal reinsurance [J].Journal of Applied Probability, 1970,7:134-156.
[12]Dufersen F,Gerber H.U.Risk theory for the compound Poisson risk model process that is perturbed by diffusion [J].Insurance:Mathematical and Economics,1991,10:51-59.
[13]Dickson C.M.D.On the distribution of the surplus prior to ruin [J].Insurance: Mathematical and Economlcs,1992,7:191-207.
[14]Dickson D.C.M,Waters H.R. Some optimal dividend problem [J].ASTIN Bulletin,2004,4:49-74.
[15]Fleming W.H,Soner H.M.Controlled markove processes and viscosity solution [M].New York:Springer Verlag,1993.
[16]Gerber H.U.Entscheidigungskriterien fur den zusammengesetzten Poisson Prozess[J].Mitteilungen der Vereinigung Schweizer Versicherungs mathematiker 1969,19:185-228.
[17]Gerber H.U.An Extension to the Renewal Equation and its Application in the Collective Theory of Risk [J].Scandinavisk Aktuarietidskrift,1970,205-210.
[18]Gerber H.U.An Introduction to Mathematical Risk Theory [M].Philadephia,S.S. 1979.
[19]Gerber H.U,Shiu E.S.W..Optimal dividends:Analysis with Brownian motion [J].North American Actuarial Journal,2004,8 (1):1-20.
[20]Grandell J.Aspects of risk theory [M].New York:Springer-Verlag,1991.
[21]Grossman S.J,Zhou Z. Optimal investment strategies for controlling drawdowns [J].Math.Finance,1993,3:241-276.
[22]Guo J,Bai L.Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint [J].Insurance Mathematical and Economlcs,2008,42:968-975.
[23]Gao J.Optimal portfolios for DC pension plans under a CEV model [J]. Insurance:Mathematical and Economlcs,2009,44:479-490
[24]Hφjgaard B,Taksar M.Controlling risk exposure and dividend pay-out scemes: Insurance company example [J].Mathematical Finance,1999,9:153-182.
[25]Hojgaard B,Taksar M.Optimal risk control for a large corporation in the presence of returns on investments [J].Finance and Stochastics, 2001,5:527-547.
[26]Hipp C,Plum M.Optimal investment for insurer [J].Insurance:Mathematical and Economlcs 2000,27:215-228.
[27]Hipp C.Optimal investment for insurer:the Coxian case [J].Prerint,2002.
[28]Hipp C,Vogt M.Optimal dynamic XL reinsurance [J].Astin Bulletin, 2003,33(2):193-207.
[29]Hu Y,Zhinming Z.The perturbed compound Poisson risk model with multilayer dividend strategy [J].Statistic and Probability Letters,2008.
[30]Hu Y,ZhinmingZ.Gerber-Shiu discounted penalty function in a Sparre Andersen model with multi-layer dividend strategy [J].Insurance:Mathematic and Economics,2008,42:984-991.
[31]Irgens C, Paulsen J. Optimal control of risk exposure, reinsurance and investments for insurance portfolios [J].Insurance:Mathematics and Economics, 2004,35:21-51.
[32]Jeanblanc-Picque, Shiryaev A.N.Optimization of the flow of dividends [J].Russian Mathematical Surveys,1999,50:257-277.
[33]Lundberg F. I. Aterforsakring av kollektivrisker. Uppsala:Almqvist and Wiksell, 1903.
[34]Li J,Wu R.Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility [J].Applied Stochastic Model in Bussiness And Industry. Published Online:6 Feb 2009.
[35]Lin X.S,Willmot G.E, Drekic S.The classical risk model with a constant dividend barrier[J].Insurance:Mathematics and Economics,2003,33:551-566.
[36]Lin X.S,Pavlova K.P. The compound Poisson risk model with a thresholds strategy [J].Insurance:Mathematic and Economics,2006,38:57-80.
[37]Lin X.S,Pavlova K.P.The compound Poisson risk model with multiple thresholds [J].Insurance:Mathematic and Economics,2008,42:617-627.
[38]Li S.The distribution of the dividend payment in the compound Poisson risk model perturbed by diffusion. Working paper,2005.
[39]Maritina T.C,Gilber P.Stochastic control theory for optimal optimal investment.School of Actuarial Studies [J].Faculty of Commerce and Economics, 2003,1-23.
[40]Paulsen J.Optimal dividend payouts for diffusions with solvency constraints [J]. Finance and Stochastics,2003,4:457-474.
[41]Paulsen J,Gjessing H.K.Optimal choice of dividend barriers for a risk process with stochastic return on investments [J].Insurance:Mathematics and Economics 1997,20:215-223.
[42]Schmidli H.On the distribution of the surplus prior to and at ruin [J].Astin Bulletin,1999,29:227-244.
[43]Schmidli H.Optimal proportional reinsurance policy in a dynamic setting [J]. Scand Actuarial J,2001,55-68.
[44]Schmidli H.On minimizing the ruin probability by investment and insurance. Technical Report,No.175,Laboratory for Actuarial Mathematical,University of Copenhagen,2002,1-24.
[45]Schmidli H.Stochastic control in insyrance [M].Springer,London,2008.
[46]Shreve S,Soner HM.Optimal investment and consumption with transaction costs [J].Annals of Applied Probability,1994,4:609-692.
[47]Shreve S,Soner HM,Xu GL.Optimal investment and consumption with two bonds and transaction costs [J].Mathematical Finance,1991,1:53-84.
[48]Sundt B,Teugels J.L.The adjustment function in ruin estimates under interest force [J].Insurance:Mathematics and Economics,1997,19:85-94.
[49]Taksar M.Optimal risk and dividend distribution control model for an insurance company [J].Math Meth Operres,2000,51:1-42.
[50]Taksar M,Hogaard.B,Asmussen,S.Optimal risk Control and Dividend Distributlon Policies.Example of Exccss-of Loss Reinsurance [J].Finance and Stochastic,2000,4:299-324.
[51]Taksar M,Asmussen S.Controlled Diffusion models for Optimal Dividend pay-out [J].Insurance:Mathematics and Economics,1997,20:1-15.
[52]Wan Ning.Dividend payment with a threshold strategy in the compound Poisson risk perturbed by diffusion [J].Insurance:Mathematics and Economics,2007,40: 509-634.
[53]Xu G.L,Shreve S.E.A duality methods for optimal consumption and investment under short-selling prohibition:Ⅱ constant market coefficients [J].The Annals of Applied Probability,1992,2:314-328.
[54]Yang H, Zhang L.Optimal investment for insurer with jump-diffusion risk process [J].Insurance:Mathematics and Economics,2005,35:21-51.
[55]Zhang Xinli,Zhang Kecun,Yu xingjiang. Optimal proportional reinsurance and investment with transaction cost maximizing the terminal wealth [J].Insurance: Mathematical and Economics,2008,44:473-478.
[56]李秀芳,曾庆五.保险精算[M].中国会融出版社,200.5.
[57]梁志彬.跳扩散模型的最优投资和再保险[J].数学学报,2008,51：1195-1204.
[58]聂高秦.金融保险中的几类风险模型[M].华中科技大学,博士学位论文,2008
[59]粟芳.非寿险精算[M].清华大学出版社,2006.
[60]王爱香.一类超额损失和比例再保险的随机控制问题[M].上海大学,硕士学位论文,2006.
[61]徐林.具有随机投资收益的风险模型下若干破产问题的研究[M].华东师范大学,博士学位论文,2008.
[62]杨步清,叶中行.保险公司的最优再保险和红利分配[J].系统工程2000,6：23-27.