伴有状态矩方程的随机最优控制变分方法及其应用
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摘要
通过对随机过程中的矩方程上应用一个非常简明的变换,能够使用Euler-lagrange变分方法解决一些随机最优控制问题。严格的说,Euler-lagrange变分方法在随机最优控制问题中并不实用,它必须借助数值函数的方法,并且在实际问题中所得到的偏微分方程并不易解。而本文通过将矩问题引入变分方法中,使得新的状态变量是原状态变量的普通样本,变化后的动态方程和目标函数可以通过变分方法,得到对一个常微分方程的解,在某些实际问题中易解。本文将这个方法应用于经济中的随机需求下商品的零售商订货策略问题,投资组合及消费选择的最优控制问题,以及再保险模型的最优控制策略问题。
在随机需求下商品的零售商订货策略问题中,结合市场需求的随机性,考虑价格的变动,以零售商期望利润最大化为出发点,寻求最优的订货策略。
在投资组合及消费选择的最优控制问题中,为使得消费和终值财富的期望效用最大化,对给定时域的投资和消费给出最优解。
在再保险模型的最优控制策略问题中,通过对一类带分红过程的比例再保险模型进行分析,采取最优控制策略。
By using a very simple remark on the moment equations of stochastic processes, one can use the Euler-Lagrange variational approach to solve some stochastic optimal control problems. In stochastic optimal control, strictly speaking, Euler-Lagrange variational calculus does not apply, and one has to use value function approach. Unfortunately, on the practical standpoint, the partial differential equation so obtained is not very manageable. In the approach, the new state variable is the deviation from the nominal trajectory. The new dynamical equation and the new cost functions may be translated into the deterministic problem of a differential equation, and it is easy to solve. In the thesis, this approach is applicated in the stochastic economics, which include the problem on the retailer's ordering policy for commodities with stochastic demand, the optimal investment and consumption problem and the optimal control policy of reinsurance model.
In the first problem,the thesis studies inventory management of commodities according to the research of stochastic demand with service income management on perishable commodities in network environment, which aims to find order quantity that maximizes the retailer's expected profit with stochastic demand.
In the second problem, the optimal investment and consumption problem is to maximize the expected-utility about the consumption and terminal wealth extreme in fixed time domail.
At last, we discuss a class of proportional reinsurance model with dividend process and also provide their optimal policies.
在随机需求下商品的零售商订货策略问题中,结合市场需求的随机性,考虑价格的变动,以零售商期望利润最大化为出发点,寻求最优的订货策略。
在投资组合及消费选择的最优控制问题中,为使得消费和终值财富的期望效用最大化,对给定时域的投资和消费给出最优解。
在再保险模型的最优控制策略问题中,通过对一类带分红过程的比例再保险模型进行分析,采取最优控制策略。
By using a very simple remark on the moment equations of stochastic processes, one can use the Euler-Lagrange variational approach to solve some stochastic optimal control problems. In stochastic optimal control, strictly speaking, Euler-Lagrange variational calculus does not apply, and one has to use value function approach. Unfortunately, on the practical standpoint, the partial differential equation so obtained is not very manageable. In the approach, the new state variable is the deviation from the nominal trajectory. The new dynamical equation and the new cost functions may be translated into the deterministic problem of a differential equation, and it is easy to solve. In the thesis, this approach is applicated in the stochastic economics, which include the problem on the retailer's ordering policy for commodities with stochastic demand, the optimal investment and consumption problem and the optimal control policy of reinsurance model.
In the first problem,the thesis studies inventory management of commodities according to the research of stochastic demand with service income management on perishable commodities in network environment, which aims to find order quantity that maximizes the retailer's expected profit with stochastic demand.
In the second problem, the optimal investment and consumption problem is to maximize the expected-utility about the consumption and terminal wealth extreme in fixed time domail.
At last, we discuss a class of proportional reinsurance model with dividend process and also provide their optimal policies.
引文
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2. S.Tang and X.Li. Necessary conditions for optimal control of Stochastic systems with random jumps [J]. SIAM J.Control and Optim.2004,Vol.32:23-45.
3. Hanzhong Wu and X.Zhou.Stochastic frequency characteristics[J].SIAM J.Control and Optim.2001,Vol.40:42-60.
4. N.Nagase. On the existence of optimal control for controlled stochastic partial dif-ferrential equations[J].Nagoya Math.J.2005,115:73-85.
5. N.Nagase,M.Nisio. Optimal controls for stochastic partial differential equations[J]. SIAM J.Control Optim.2003,28(1):186-213.
6.N.El Karoui, D.Huu Nguyen,M. Jeanblanc-Pique. Compactilcation methods in the control of degenerate diffusions[J].Stochastics,1998,20:169-219.
7. M.Nisio. Optimal control for stochastic partial differential equations and viscosity solutions of Bellman equations [J]. Nagoya Math.J.2000.123:13-37.
8. M.Nisio. On sensitive control for stochastic partial differential equations, in: H.Kunitaetal, Stchastic Analysis on Inlnite Dimensional Spaces, Proceedings of the U.S-Japan Bilateral Seminar,1994,Vol.310:231-241.
9. M.Nisio, Game approach to risk sensitive control for stochastic evolution systems, in:W.M. Mceneaney, etal. Stochastic Analysis, Control, Optimization and Applica-tions, A volume in Honor of Wendell H. Fleming, on the Occasion of his 70th Birthday, Birkhauser, Boston,1999,115-134.
10. D.Gatarek, J.Sobczyk. On the existence of optimal controls of Hilbert space valued diffusions [J]. SIAM J. Control Optim.1994,32(1):170-175.
11. G.Daprato, S.Kwapien,J.Zabczyk. Regularity of solutions of linear stochastic equa-tions in Hilbert spaces [J]. Stochastics,1987,23:1-23.
12. Rainer Buckdahn, Aurel Rascanu. On the existence of stochastic optimal control of distributed state system [J]. Nolinear Analysis,2003,52:1153-1184.
13. Fleming WH,Soner HM.Controlled Markov Processes and Viscosity Solutions [M]. Berlin, Heidelberg, New Yory:Springer-Verlag,2000.281-313.
14. Jerome L. Stein. Applications of stochastic optimal control dynamic programming to international finance and debt crises[J]. Nonlinear Analysis,2005,63:2033-2041.
15. A.H.Jazwinski, "Stochastic Processes and Filtering Theory"[M], New York, Acad-emic Press,1995.
16. Schachermayer, W.Wang, H.Utility maximization in incomplete markets with ran-dom endowmen[J], Finance stochast,2001,5:259-272.
17. Browne S.Optimal investment polices for a firm with a random risk process:Expo-nential utility and minimizing the probability of ruin [J]. Mathematics of Opera-tions Research,1999,20(4):937-958.
18. Shreve SE, Lehoczky J P, Gaver D P.Optimal consumption for general diffusions with absorbing and reflecting barrers [J]. SIAM on Control and optimization,2003, 22(1):55-75.
19. Kabanov,Y.M.,Stricker,c.,On the optimal portfolio for the exponential utility Max-imization:Remarks to the six—author paper [J],Math Finance,2002,12,125-134.
20. Merton,R.Optimal consumption and portfolio rules in a continuous-time model[J], Journal of Economic Theory,1991,3:373-413.
21. Kolatzas, I.Lehoczky,J.and shreve,s., Optimal portfolio and consumption decisions or a small investor on a finite horizon[J], SIAM Journal on control and optimization,1997,25:1557-1586.
22.周庆健,张魁元,吴建民,线性加指数效用函数的组合投资研究[J],应用数学,2004,17：53-58。
23.严士键,王隽骧,刘秀芳,概率论基础[M]北京：科学出版社,1999,164-178。
24. Liu Kunhui.The existence and uniqueness of solution for a class of stochastic func-tional equations on S.P.space [J].Acta Mathematica Sinica,2001,21 (3):391-400.
25.郭子君,吴让泉。具有异常波动市场的消费与投资策略[M]。控制理论与应用,2004,8：546-547。
26.史敬涛,吴臻。一类证券市场中投资组合及消费选择的最优控制问题。应用数学报,2005,20(1)：1-8。
27. R.C. Merton, Optimal consumption and portfolio rules in a continuous-time model [J]. Economic Theory,1992,Vol.3,pp.373-413.
28. Merton,R.Lifetime portfolio selection under uncertainty:the continuoustime case [J].Review of economics and statistics[J],1969,51:247-257.
29.荣喜民,张世英。再保险定价的研究[J]。系统工程学报,2001,16(6)：471-474。
30. Tomasz R,Hanspeter S, Volker S, et al. Stochastic Processes for Insurance and Fin-ance[M]. Chicester;New York; J.Wiley,1998.14-17.
31. Hogaard B,Taksar M.Optimal proportional reinsurance policies for diffusion mod-els with tansaction costs [J]. Insurance; Mathematics and Economics,1998,22(1): 41-51.
32. Paulsen J, Gjessing H K.Optimal choice of dividend barriers for a risk prociss with stochastic return of investment [J]. Insurance; Mathematics and Economics,1997, 20(3):215-223.
2. S.Tang and X.Li. Necessary conditions for optimal control of Stochastic systems with random jumps [J]. SIAM J.Control and Optim.2004,Vol.32:23-45.
3. Hanzhong Wu and X.Zhou.Stochastic frequency characteristics[J].SIAM J.Control and Optim.2001,Vol.40:42-60.
4. N.Nagase. On the existence of optimal control for controlled stochastic partial dif-ferrential equations[J].Nagoya Math.J.2005,115:73-85.
5. N.Nagase,M.Nisio. Optimal controls for stochastic partial differential equations[J]. SIAM J.Control Optim.2003,28(1):186-213.
6.N.El Karoui, D.Huu Nguyen,M. Jeanblanc-Pique. Compactilcation methods in the control of degenerate diffusions[J].Stochastics,1998,20:169-219.
7. M.Nisio. Optimal control for stochastic partial differential equations and viscosity solutions of Bellman equations [J]. Nagoya Math.J.2000.123:13-37.
8. M.Nisio. On sensitive control for stochastic partial differential equations, in: H.Kunitaetal, Stchastic Analysis on Inlnite Dimensional Spaces, Proceedings of the U.S-Japan Bilateral Seminar,1994,Vol.310:231-241.
9. M.Nisio, Game approach to risk sensitive control for stochastic evolution systems, in:W.M. Mceneaney, etal. Stochastic Analysis, Control, Optimization and Applica-tions, A volume in Honor of Wendell H. Fleming, on the Occasion of his 70th Birthday, Birkhauser, Boston,1999,115-134.
10. D.Gatarek, J.Sobczyk. On the existence of optimal controls of Hilbert space valued diffusions [J]. SIAM J. Control Optim.1994,32(1):170-175.
11. G.Daprato, S.Kwapien,J.Zabczyk. Regularity of solutions of linear stochastic equa-tions in Hilbert spaces [J]. Stochastics,1987,23:1-23.
12. Rainer Buckdahn, Aurel Rascanu. On the existence of stochastic optimal control of distributed state system [J]. Nolinear Analysis,2003,52:1153-1184.
13. Fleming WH,Soner HM.Controlled Markov Processes and Viscosity Solutions [M]. Berlin, Heidelberg, New Yory:Springer-Verlag,2000.281-313.
14. Jerome L. Stein. Applications of stochastic optimal control dynamic programming to international finance and debt crises[J]. Nonlinear Analysis,2005,63:2033-2041.
15. A.H.Jazwinski, "Stochastic Processes and Filtering Theory"[M], New York, Acad-emic Press,1995.
16. Schachermayer, W.Wang, H.Utility maximization in incomplete markets with ran-dom endowmen[J], Finance stochast,2001,5:259-272.
17. Browne S.Optimal investment polices for a firm with a random risk process:Expo-nential utility and minimizing the probability of ruin [J]. Mathematics of Opera-tions Research,1999,20(4):937-958.
18. Shreve SE, Lehoczky J P, Gaver D P.Optimal consumption for general diffusions with absorbing and reflecting barrers [J]. SIAM on Control and optimization,2003, 22(1):55-75.
19. Kabanov,Y.M.,Stricker,c.,On the optimal portfolio for the exponential utility Max-imization:Remarks to the six—author paper [J],Math Finance,2002,12,125-134.
20. Merton,R.Optimal consumption and portfolio rules in a continuous-time model[J], Journal of Economic Theory,1991,3:373-413.
21. Kolatzas, I.Lehoczky,J.and shreve,s., Optimal portfolio and consumption decisions or a small investor on a finite horizon[J], SIAM Journal on control and optimization,1997,25:1557-1586.
22.周庆健,张魁元,吴建民,线性加指数效用函数的组合投资研究[J],应用数学,2004,17：53-58。
23.严士键,王隽骧,刘秀芳,概率论基础[M]北京：科学出版社,1999,164-178。
24. Liu Kunhui.The existence and uniqueness of solution for a class of stochastic func-tional equations on S.P.space [J].Acta Mathematica Sinica,2001,21 (3):391-400.
25.郭子君,吴让泉。具有异常波动市场的消费与投资策略[M]。控制理论与应用,2004,8：546-547。
26.史敬涛,吴臻。一类证券市场中投资组合及消费选择的最优控制问题。应用数学报,2005,20(1)：1-8。
27. R.C. Merton, Optimal consumption and portfolio rules in a continuous-time model [J]. Economic Theory,1992,Vol.3,pp.373-413.
28. Merton,R.Lifetime portfolio selection under uncertainty:the continuoustime case [J].Review of economics and statistics[J],1969,51:247-257.
29.荣喜民,张世英。再保险定价的研究[J]。系统工程学报,2001,16(6)：471-474。
30. Tomasz R,Hanspeter S, Volker S, et al. Stochastic Processes for Insurance and Fin-ance[M]. Chicester;New York; J.Wiley,1998.14-17.
31. Hogaard B,Taksar M.Optimal proportional reinsurance policies for diffusion mod-els with tansaction costs [J]. Insurance; Mathematics and Economics,1998,22(1): 41-51.
32. Paulsen J, Gjessing H K.Optimal choice of dividend barriers for a risk prociss with stochastic return of investment [J]. Insurance; Mathematics and Economics,1997, 20(3):215-223.