具有模型不确定性的最优投资和再保险策略
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摘要
近年来,由于保险行业竞争激烈,保险公司一方面通过对公司盈余进行投资,从投资中获得大量的收益来提高自己的偿付能力,同时保险公司为了减少自身所面临的大赔付的风险,又必须对赔付进行再保险处理。控制投资和再保险,使得期望财富效用最大或破产概率最小,无论在理论上,还是在保险实务中,都具有十分重要的意义。
本文首先对进行停止损失再保险的传统风险模型做扩散逼近,使模型简单化,以最小破产概率为目标函数,借助HJB方程,求得了最优自留额满足的表达式,在个体理赔取指数分布和Erlang(2)分布的情况下求出了最优自留额和最小破产概率的显示解,并通过数值计算得到自留额与各参数之间的关系。其次,对不确定情况下的跳-扩散风险最优投资和再保险进行研究。在保险公司的盈余和投资具有不确定性的条件下,得到了使得期望指数效用最大的最优投资、再保险和概率测度变换,以及最优值函数的显示表达式。
In recent year, due to severe competition in the insurance industry, the insurance companies get a lot of benefits to improve their solvency from the investment of the money at its disposal. At the same time, in order to reduce the risk of large claims, the insurance company takes reinsurance policy for claims. Subject to the control of investment and reinsurance, maximizing the expected wealth utility and minimizing the probability of ruin have a great deal of theoretical and practical significance.
Firstly, the paper studies the diffusion approximation of the classical risk model. The HJB equation of the minimal probability of ruin and the optimal stop-loss reinsurance is given. We obtain the explicit expression of the minimal probability of ruin and the optimal stop-loss reinsurance under the claim is exponential distribution and Erlang distribution. The effects of the parameters on the optimal retention level are obtained by numerical calculation. Secondly, we study the optimal investment and reinsurance in the jump-diffusion risk model under uncertainty. Under uncertainty of the surplus and investment, explicit expressions for the maximal expected exponential utility and the corresponding optimal policies are obtained.
本文首先对进行停止损失再保险的传统风险模型做扩散逼近,使模型简单化,以最小破产概率为目标函数,借助HJB方程,求得了最优自留额满足的表达式,在个体理赔取指数分布和Erlang(2)分布的情况下求出了最优自留额和最小破产概率的显示解,并通过数值计算得到自留额与各参数之间的关系。其次,对不确定情况下的跳-扩散风险最优投资和再保险进行研究。在保险公司的盈余和投资具有不确定性的条件下,得到了使得期望指数效用最大的最优投资、再保险和概率测度变换,以及最优值函数的显示表达式。
In recent year, due to severe competition in the insurance industry, the insurance companies get a lot of benefits to improve their solvency from the investment of the money at its disposal. At the same time, in order to reduce the risk of large claims, the insurance company takes reinsurance policy for claims. Subject to the control of investment and reinsurance, maximizing the expected wealth utility and minimizing the probability of ruin have a great deal of theoretical and practical significance.
Firstly, the paper studies the diffusion approximation of the classical risk model. The HJB equation of the minimal probability of ruin and the optimal stop-loss reinsurance is given. We obtain the explicit expression of the minimal probability of ruin and the optimal stop-loss reinsurance under the claim is exponential distribution and Erlang distribution. The effects of the parameters on the optimal retention level are obtained by numerical calculation. Secondly, we study the optimal investment and reinsurance in the jump-diffusion risk model under uncertainty. Under uncertainty of the surplus and investment, explicit expressions for the maximal expected exponential utility and the corresponding optimal policies are obtained.
引文
[1]Christian I., Jostein P. Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insurance:Mathematics and Economics, 2004, (35):21-51
[2]李秀芳,曾庆五.《保险精算》.中国金融出版社,2005
[3]粟芳.《非寿险精算》.清华大学出版社,2006
[4]Gerber H.U. The dilemma between dividends and safety and a generalization of the Lundeberg-Cramer formulas. Scand. Actuarial J.,1974,46-57
[5]Grandits P. An analogue of the Cramer-Lundberg approximation in the optimal investment case. Appl. Math.Optim.2004, (50):1-20
[6]Browne S. Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Mathematics Methods Operator Research,1995, (20):937-957
[7]Browne S. Survival and growth with liability:optimal portfolio strategies in continuous time. Mathematics Methods Operator Research,1997, (22):468-493
[8]Browne S. Beating a moving target:optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastic,1999, (3):275-294
[9]Liang Z.B. Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion. Acta Mathematicae Applicatae Sinica, English Series, 2007,23(3):477-488
[10]Liang Z.B., Guo J.Y. Upper bound for ruin probabilities under optimal investment and proportional reinsurance. Appl.Stochastic Models Bus. Ind., 2008,(24):109-128
[11]Liang Z.B., Guo J.Y. Optimal proportional reinsurance and ruin probability. Stochastic Models,2007,23(2):333-350
[12]Liang Z. Optimal investment and Reinsurance for the Jump-Diffusion Surplus Process. Acta mathematics sinica, Chinese Series,2008, (51):1195-1204
[13]Schmidli H. Optimal proportional reinsurance policy in a dynamic setting. Scandinavian Actuarial Journal,2001, (1):55-68
[14]Promislow S.D., Young V.R. Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal,2005, 9(3),109-128
[15]Taksar M., Markussen C. Optimal dynamic reinsurance policies for large insurance portfolios. Finance Schochast,2003. (7):97-121
[16]Luo S.Z., Taksar M., Tsoi A. On reinsurance and investment for large insurance portfolios. Insurance:Mathematics and Economics,2008, (42):434-44
[17]Hipp C., Plum M. Optimal investment for insurers. Insurance:Mathematics and Economics.,2000, (27):215-228
[18]Hipp C., Plum M. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastic,2003, (7):299-321
[19]Lin C., Yang H. Optimal investment for an insurer to minimize its probability of ruin. North American Actuarial Journal,2004,8(2):11-31
[20]Yang H., Zhang L. Martingale method for ruin probability in an autoregressive model with constant interest rate. Probability in the Engineering and informational Sciences,2003, (17):183-198
[21]Yang H, Zhang L. On the distribution of surplus immediately after ruin under interest. Insurance:Mathematics and Economics,2001, (29):247-255
[22]Schmidli H. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability,2002, (12):890-907
[23]Lin X. Ruin theory for classical risk process that is perturbed by diffusion with risky investments. Appl.Stochastic Models Bus.Ind,2009, (25):33-44
[24]Yang H, Zhang L. Optimal investment for insurer with jump-diffusion risk process. Insurance:Mathematics and Economics,2005, (35):21-51
[25]Bai L.H., Guo J.Y. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance:Mathematics and Economics,2008, (42):968-975
[26]Li J., Wu R.Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility.Applied Stochastic Model in Bussiness and Industry. Published Online:6 Feb 2009
[27]Cairns A. A discussion of parameter and model uncertainty in insurance. Insurance:Mathematics and Economics,2000,27(3):313-330
[28]Cont R. Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance,2006,16(3):519-547
[29]赵巍；何建敏.基于测度变换方法的随机型创新幂式期权定价.中国管理科学,2009,(3)：34-39
[30]钱晓松.跳扩散模型中的测度变换与期权定价.应用概率统计,2004
[31]王伟,王文胜,王帅.跳扩散模型中远期生效看涨期权的定价.华东师范大学学报(自然科学版),2009,(5)：107-117
[32]Zhang X., Siu T.K.. Optimal investment and reinsurance of an insurer with model uncertainty. Insurance:Mathematics and Economics,2009, (45):81-88
[33]Hald M., Schmidli H. On the maximization of the adjustment coefficient under proportional reinsurance. Astin bull,2004, (34):75-83
[34]王爱香.《一类超额损失和比例再保险的随机控制问题》.上海大学,硕士学位论文,2006
[35]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》.华东师范大学,博士学位论文,2008
[36]Liu C.S., Yang H.Y. Optimal investment for an insurer to minimizing its probability of ruin. North American Actuarial Journal,2004, (2):11-31
[37]Shreve S, Soner HM, Xu GL. Optimal investment and consumption with two bonds and transaction costs. Mathematical Finance,1991, (1):53-84
[38]Sundt B, Teugels J.L. The adjustment function in ruin estimates under interest force. Insurance:Mathematics and Economics,1997, (19):85-94
[39]Xu G.L, Shreve S.E. A duality method for optimal consumption and investment under short-selling prohibition:II constant market coefficients. The Annals of Applied Probability,1992, (2):314-328
[40]Delbaen F., Haezendonck J.A. Martingale approach to premium calculation principles in an arbitrage free market. Insurance:Mathematics and Economics, 1989,(8)：269-277
[41]聂高秦.《金融保险中的几类风险模型》.华中科技大学,博士学位论文,2008
[42]梁志彬.跳扩散模型的最优投资和再保险.数学学报,2008,(51)：1195-1204
[43]Bellman R E. Dynamical Programming: Princeton Univ Pressing,1957
[44]Derman E. Model risk. Quantitative Strategies Research Notes,1996
[45]Dickson C.M.D. On the distribution of the surplus prior to ruin. Insurance: Mathematics and Economlcs,1992, (7):191-207
[46]Gerber H.U. Martingales in risk theory. Schweiz.Verein. Versicherungsmath.Mitt.1973,(73):205-216
[47]Grandits P. Minimal ruin probabilities and investment under interest force for a class of subexponential distributions. Scand. Actuarial J.2005,401-416
[48]Gerber H.U. An Introduction to Mathematical Risk Theory. S.S.Heubner Foundation Monographs, University of Pennsilvania.1979
[49]He L., Hou P., Liang Z. Optimal control of the insurance company with proportional reinsurance policy under solvency constraints. Insurance: Mathematics and Economics, doi:10.1016/j. insuratheco.2008,09.004
[50]刘家有,刘再明.保险公司在固定利率下的离散型破产概率.数学理论与应用,Vol.2004,(24)：101-104
[51]张波,代金.经济环境下引入投资的古典风险模型的破产概率.经济数学,2005,(22)：111-117
[52]张连增.再保险对破产概率的影响.数量经济技术经济研究,1999,(6)：59-62
[53]Wang G.J., Wu, R. Some distribution for classical risk process that is per-turbed by diffusion. Insurance:Mathematics and Economics,2000, (26):15-24
[54]何树红,夏梓祥.再保险对时间盈余风险模型破产概率的影响.云南大学学报(自然科学版)(增刊),2004(26)：1-4
[55]Fleming W.H, Soner H.M. Controlled markove processes and viscosity solution. New York:Springer Verlag,1993
[56]Paulsen J. Optimal dividend payouts for diffusions with solvency constraints. Insurance:Finance and Stochastics,2003, (4):457-474
[57]Paulsen J, Gjessing H.K. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance:Mathematics and Economics 1997, (20),215-223
[58]Paulsen J., Gjessing H.K. Ruin theory with stochastic return on investments, Advil Appl. Probab,1997, (29):965-985
[59]Dickson D.C.M, Waters H.R. Some optimal dividend problem. ASTIN Bulletin, 2004, (4):49-74
[60]Qian Y P, Lin X. Ruin probabilities under optimal investment and proportional reinsurance policy in jump diffusion risk process. The ANZIAM Journal,2009
[2]李秀芳,曾庆五.《保险精算》.中国金融出版社,2005
[3]粟芳.《非寿险精算》.清华大学出版社,2006
[4]Gerber H.U. The dilemma between dividends and safety and a generalization of the Lundeberg-Cramer formulas. Scand. Actuarial J.,1974,46-57
[5]Grandits P. An analogue of the Cramer-Lundberg approximation in the optimal investment case. Appl. Math.Optim.2004, (50):1-20
[6]Browne S. Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Mathematics Methods Operator Research,1995, (20):937-957
[7]Browne S. Survival and growth with liability:optimal portfolio strategies in continuous time. Mathematics Methods Operator Research,1997, (22):468-493
[8]Browne S. Beating a moving target:optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastic,1999, (3):275-294
[9]Liang Z.B. Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion. Acta Mathematicae Applicatae Sinica, English Series, 2007,23(3):477-488
[10]Liang Z.B., Guo J.Y. Upper bound for ruin probabilities under optimal investment and proportional reinsurance. Appl.Stochastic Models Bus. Ind., 2008,(24):109-128
[11]Liang Z.B., Guo J.Y. Optimal proportional reinsurance and ruin probability. Stochastic Models,2007,23(2):333-350
[12]Liang Z. Optimal investment and Reinsurance for the Jump-Diffusion Surplus Process. Acta mathematics sinica, Chinese Series,2008, (51):1195-1204
[13]Schmidli H. Optimal proportional reinsurance policy in a dynamic setting. Scandinavian Actuarial Journal,2001, (1):55-68
[14]Promislow S.D., Young V.R. Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal,2005, 9(3),109-128
[15]Taksar M., Markussen C. Optimal dynamic reinsurance policies for large insurance portfolios. Finance Schochast,2003. (7):97-121
[16]Luo S.Z., Taksar M., Tsoi A. On reinsurance and investment for large insurance portfolios. Insurance:Mathematics and Economics,2008, (42):434-44
[17]Hipp C., Plum M. Optimal investment for insurers. Insurance:Mathematics and Economics.,2000, (27):215-228
[18]Hipp C., Plum M. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastic,2003, (7):299-321
[19]Lin C., Yang H. Optimal investment for an insurer to minimize its probability of ruin. North American Actuarial Journal,2004,8(2):11-31
[20]Yang H., Zhang L. Martingale method for ruin probability in an autoregressive model with constant interest rate. Probability in the Engineering and informational Sciences,2003, (17):183-198
[21]Yang H, Zhang L. On the distribution of surplus immediately after ruin under interest. Insurance:Mathematics and Economics,2001, (29):247-255
[22]Schmidli H. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability,2002, (12):890-907
[23]Lin X. Ruin theory for classical risk process that is perturbed by diffusion with risky investments. Appl.Stochastic Models Bus.Ind,2009, (25):33-44
[24]Yang H, Zhang L. Optimal investment for insurer with jump-diffusion risk process. Insurance:Mathematics and Economics,2005, (35):21-51
[25]Bai L.H., Guo J.Y. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance:Mathematics and Economics,2008, (42):968-975
[26]Li J., Wu R.Optimal investment problem with stochastic interest rate and stochastic volatility:maximizing a power utility.Applied Stochastic Model in Bussiness and Industry. Published Online:6 Feb 2009
[27]Cairns A. A discussion of parameter and model uncertainty in insurance. Insurance:Mathematics and Economics,2000,27(3):313-330
[28]Cont R. Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance,2006,16(3):519-547
[29]赵巍；何建敏.基于测度变换方法的随机型创新幂式期权定价.中国管理科学,2009,(3)：34-39
[30]钱晓松.跳扩散模型中的测度变换与期权定价.应用概率统计,2004
[31]王伟,王文胜,王帅.跳扩散模型中远期生效看涨期权的定价.华东师范大学学报(自然科学版),2009,(5)：107-117
[32]Zhang X., Siu T.K.. Optimal investment and reinsurance of an insurer with model uncertainty. Insurance:Mathematics and Economics,2009, (45):81-88
[33]Hald M., Schmidli H. On the maximization of the adjustment coefficient under proportional reinsurance. Astin bull,2004, (34):75-83
[34]王爱香.《一类超额损失和比例再保险的随机控制问题》.上海大学,硕士学位论文,2006
[35]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》.华东师范大学,博士学位论文,2008
[36]Liu C.S., Yang H.Y. Optimal investment for an insurer to minimizing its probability of ruin. North American Actuarial Journal,2004, (2):11-31
[37]Shreve S, Soner HM, Xu GL. Optimal investment and consumption with two bonds and transaction costs. Mathematical Finance,1991, (1):53-84
[38]Sundt B, Teugels J.L. The adjustment function in ruin estimates under interest force. Insurance:Mathematics and Economics,1997, (19):85-94
[39]Xu G.L, Shreve S.E. A duality method for optimal consumption and investment under short-selling prohibition:II constant market coefficients. The Annals of Applied Probability,1992, (2):314-328
[40]Delbaen F., Haezendonck J.A. Martingale approach to premium calculation principles in an arbitrage free market. Insurance:Mathematics and Economics, 1989,(8)：269-277
[41]聂高秦.《金融保险中的几类风险模型》.华中科技大学,博士学位论文,2008
[42]梁志彬.跳扩散模型的最优投资和再保险.数学学报,2008,(51)：1195-1204
[43]Bellman R E. Dynamical Programming: Princeton Univ Pressing,1957
[44]Derman E. Model risk. Quantitative Strategies Research Notes,1996
[45]Dickson C.M.D. On the distribution of the surplus prior to ruin. Insurance: Mathematics and Economlcs,1992, (7):191-207
[46]Gerber H.U. Martingales in risk theory. Schweiz.Verein. Versicherungsmath.Mitt.1973,(73):205-216
[47]Grandits P. Minimal ruin probabilities and investment under interest force for a class of subexponential distributions. Scand. Actuarial J.2005,401-416
[48]Gerber H.U. An Introduction to Mathematical Risk Theory. S.S.Heubner Foundation Monographs, University of Pennsilvania.1979
[49]He L., Hou P., Liang Z. Optimal control of the insurance company with proportional reinsurance policy under solvency constraints. Insurance: Mathematics and Economics, doi:10.1016/j. insuratheco.2008,09.004
[50]刘家有,刘再明.保险公司在固定利率下的离散型破产概率.数学理论与应用,Vol.2004,(24)：101-104
[51]张波,代金.经济环境下引入投资的古典风险模型的破产概率.经济数学,2005,(22)：111-117
[52]张连增.再保险对破产概率的影响.数量经济技术经济研究,1999,(6)：59-62
[53]Wang G.J., Wu, R. Some distribution for classical risk process that is per-turbed by diffusion. Insurance:Mathematics and Economics,2000, (26):15-24
[54]何树红,夏梓祥.再保险对时间盈余风险模型破产概率的影响.云南大学学报(自然科学版)(增刊),2004(26)：1-4
[55]Fleming W.H, Soner H.M. Controlled markove processes and viscosity solution. New York:Springer Verlag,1993
[56]Paulsen J. Optimal dividend payouts for diffusions with solvency constraints. Insurance:Finance and Stochastics,2003, (4):457-474
[57]Paulsen J, Gjessing H.K. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance:Mathematics and Economics 1997, (20),215-223
[58]Paulsen J., Gjessing H.K. Ruin theory with stochastic return on investments, Advil Appl. Probab,1997, (29):965-985
[59]Dickson D.C.M, Waters H.R. Some optimal dividend problem. ASTIN Bulletin, 2004, (4):49-74
[60]Qian Y P, Lin X. Ruin probabilities under optimal investment and proportional reinsurance policy in jump diffusion risk process. The ANZIAM Journal,2009