倒向随机微分方程的最优控制，微分对策和熵风险约束下的最优投资
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摘要
本文研究倒向随机微分方程的转换和停止混合最优控制,二人零和微分对策以及熵风险约束下的最优投资.全文分为两部分。
第一部分讨论倒向随机微分方程的转换和停止混合最优控制和二人零和微分对策。在第二章,我们把项目的转换和停止看作实物期权进行数学建模和求解。在第三章和第四章,我们把第二章的问题扩展为一般的倒向随机微分方程的转换和停止。倒向方程的生成元和终端值允许在给定的有限个模式之间转换,并且每次转换都需要一个正的花费。根据转换者和停止者的目标是否一致,我们讨论了混合最优控制以及零和微分对策,给出了最优控制,证明了微分对策值过程的存在性。这两类问题的HJB方程和HJBI方程分别是具有单侧和双侧混合障碍的多维反射倒向随机微分方程。对于具有单侧混合障碍的多维反射倒向随机微分方程,我们用惩罚方法证明了适应解的存在性;通过将解的第一部分表示为混合最优控制问题的值过程,我们得到了解的唯一性。对于具有双侧混合障碍的多维反射倒向随机微分方程,借助于广义单调极限定理,我们用Picard迭代方法证明了适应解的存在性;通过将解的第一部分表示为一维反射倒向随机微分方程的最优转换问题的值过程,我们得到了解的唯一性。第五章介绍在略微不同的假定条件下如何用惩罚方法来证明具有双侧混合障碍的多维反射倒向随机微分方程解的存在性。
第二部分讨论熵风险约束下的最优投资。在第六章,我们首先介绍一致风险度量和凸风险度量的概念,然后推广熵风险度量的定义,最后研究熵风险度量所具有的性质。在第七章,我们用熵风险度量来描述风险约束,讨论投资组合的最优选择。通过分析对偶问题,我们得到了原问题解的存在唯一性,给出了最优终端财富的表达式。
This thesis investigates the mixed optimal control and the zero-sum differential game on the switching and stopping of backward stochastic differential equations (BSDEs,for short),and the optimal investment with an entropic risk constraint.It consists of two parts.
Part 1 is concerned with the mixed optimal control and the zero-sum differential game on the switching and stopping of BSDEs.It consists of Chapters 2,3 and 4. In Chapter 2,we model and solve the switching and stopping of a project as a real option.In Chapters 3 and 4,the problem discussed in Chapter 2 is generalized to the switching and stopping of general BSDEs whose generator and terminal value are allowed to be switched among a finite number of given modes with a positive cost being incurred for each switching.By distinguishing the coincidence and the contradiction between the switcher's and the stopper's purpose,we study the mixed optimal control and the zero-sum differential game therein.The optimal control is obtained for the former problem and the existence of the value process is proved for the latter.The respective HJB equation and HJBI equation of both problems turn out to be multi-dimensional reflected BSDEs with one-sided and two-sided hybrid barriers.For the multi-dimensional reflected BSDEs with one-sided hybrid barriers, we prove the existence of the solution by the penalization method,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the mixed optimal control problem.For the multi-dimensional reflected BSDEs with two-sided hybrid barriers,we prove the existence of the solution by the Picard iteration method,invoking the generalized monotonic limit theorem,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the optimal switching problem of one-dimensional reflected BSDEs.In Chapter 5,the existence of the solution to the multi-dimensional reflected BSDEs with two-sided hybrid barriers is further proved by a penalization method under some slight different assumptions.
Part 2 is concerned with the optimal investment with a constraint on the entropic risk for the terminal wealth.It consists of Chapters 6 and 7 In Chapter 6,we introduce the concepts of coherent risk measure and convex risk measure, then generalize the definition of entropic risk measure and examine the properties of entropic risk measure.In Chapter 7,we study the optimal portfolio selection with a risk constraint which is expressed in terms of the entropie risk measure.By examining the dual problem,we prove the existence and uniqueness of the solution, and obtain an expression for the optimal terminal wealth profile.
第一部分讨论倒向随机微分方程的转换和停止混合最优控制和二人零和微分对策。在第二章,我们把项目的转换和停止看作实物期权进行数学建模和求解。在第三章和第四章,我们把第二章的问题扩展为一般的倒向随机微分方程的转换和停止。倒向方程的生成元和终端值允许在给定的有限个模式之间转换,并且每次转换都需要一个正的花费。根据转换者和停止者的目标是否一致,我们讨论了混合最优控制以及零和微分对策,给出了最优控制,证明了微分对策值过程的存在性。这两类问题的HJB方程和HJBI方程分别是具有单侧和双侧混合障碍的多维反射倒向随机微分方程。对于具有单侧混合障碍的多维反射倒向随机微分方程,我们用惩罚方法证明了适应解的存在性;通过将解的第一部分表示为混合最优控制问题的值过程,我们得到了解的唯一性。对于具有双侧混合障碍的多维反射倒向随机微分方程,借助于广义单调极限定理,我们用Picard迭代方法证明了适应解的存在性;通过将解的第一部分表示为一维反射倒向随机微分方程的最优转换问题的值过程,我们得到了解的唯一性。第五章介绍在略微不同的假定条件下如何用惩罚方法来证明具有双侧混合障碍的多维反射倒向随机微分方程解的存在性。
第二部分讨论熵风险约束下的最优投资。在第六章,我们首先介绍一致风险度量和凸风险度量的概念,然后推广熵风险度量的定义,最后研究熵风险度量所具有的性质。在第七章,我们用熵风险度量来描述风险约束,讨论投资组合的最优选择。通过分析对偶问题,我们得到了原问题解的存在唯一性,给出了最优终端财富的表达式。
This thesis investigates the mixed optimal control and the zero-sum differential game on the switching and stopping of backward stochastic differential equations (BSDEs,for short),and the optimal investment with an entropic risk constraint.It consists of two parts.
Part 1 is concerned with the mixed optimal control and the zero-sum differential game on the switching and stopping of BSDEs.It consists of Chapters 2,3 and 4. In Chapter 2,we model and solve the switching and stopping of a project as a real option.In Chapters 3 and 4,the problem discussed in Chapter 2 is generalized to the switching and stopping of general BSDEs whose generator and terminal value are allowed to be switched among a finite number of given modes with a positive cost being incurred for each switching.By distinguishing the coincidence and the contradiction between the switcher's and the stopper's purpose,we study the mixed optimal control and the zero-sum differential game therein.The optimal control is obtained for the former problem and the existence of the value process is proved for the latter.The respective HJB equation and HJBI equation of both problems turn out to be multi-dimensional reflected BSDEs with one-sided and two-sided hybrid barriers.For the multi-dimensional reflected BSDEs with one-sided hybrid barriers, we prove the existence of the solution by the penalization method,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the mixed optimal control problem.For the multi-dimensional reflected BSDEs with two-sided hybrid barriers,we prove the existence of the solution by the Picard iteration method,invoking the generalized monotonic limit theorem,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the optimal switching problem of one-dimensional reflected BSDEs.In Chapter 5,the existence of the solution to the multi-dimensional reflected BSDEs with two-sided hybrid barriers is further proved by a penalization method under some slight different assumptions.
Part 2 is concerned with the optimal investment with a constraint on the entropic risk for the terminal wealth.It consists of Chapters 6 and 7 In Chapter 6,we introduce the concepts of coherent risk measure and convex risk measure, then generalize the definition of entropic risk measure and examine the properties of entropic risk measure.In Chapter 7,we study the optimal portfolio selection with a risk constraint which is expressed in terms of the entropie risk measure.By examining the dual problem,we prove the existence and uniqueness of the solution, and obtain an expression for the optimal terminal wealth profile.
引文
[1]Alexander,G.J.and A.M.Baptista(2004):A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model,Management Science 50(9),1261-1273.
[2]Artzner,P.,F.Delbaen and J.M.Eber,D.Heath(1999):Coherent measures of risk,Mathematical Finance 9(3),203-228.
[3]Barrieu,P.and N.El Karoui(2004):Optimal Derivatives Design under Dynamic Risk Measures,Contemporary Mathematics,A.M.S.Proceedings.
[4]Basak,S.and A.Shapiro(2001):Value-at-Risk-Based Risk Management:Optimal Policies and Asset Prices,The review of financial studies summer 14(2),371-405.
[5]Benati,S.(2003):The optimal portfolio problem with coherent risk measure constraints,European Journal of Operational Research 150,572-584.
[6]Bensoussan,A.and J.L.Lions(1984):Impulse control and quasivariational inequalities,Ganthier-Villars,Montrouge.
[7]Brennan,M.J.and E.S.Schwartz(1985):Evaluating natural resource investments,J.Business 58,135-157.
[8]Briand,P.and Y.Shapiro(2006):BSDE with quadratic growth and unbounded terminal value,Probab.Theory Relat.Fields 136,604-618.
[9]Carmona,R.and M.Ludkovski(2005):Optimal Switching with Applications to Energy Tolling Agreements,preprint.
[10]Chen Z.and L.Epstein(2002):Ambiguity,risk and asset returns in continuous time Econometrica 70(4),1403-1443.
[11]Cvitanic,J.and I.Karatzas(1996):Backward SDEs with reflection and Dynkin games,Annals of Probability 24(4),2024-2056
[12]Delbaen,F.(2002):Coherent risk measures on general probability spaces,in Sandmann,K.(ed.) et al.,Advances in Finance and Stochastics,1-37,Springer-Verlag.
[13]Dellacherie,C.and P.A.Meyer(1980):Probabilit(?)s et Potentiel,Ⅰ-Ⅳ.Hermann,Paris.
[14]Detlefsen,K.and G.Scandolo(2005):Conditional and dynamic convex risk measures,Finance and Stochastics 9(4),539-561.
[15]Dixit,A.(1989):Entry and exit decisions under uncertainty,J.Political Economy 97,620-638.
[16]Dixit,A.and R.S.Pindyck(1994):Investment under uncertainty,Princeton Univ.Press.Princeton,NJ.
[17]Djehiche,B.and S.Hamad(?)ne(2007):On a finite horizon starting and stopping problem with default risk,preprint.
[18]Duckworth,K.and M.Zervos(2000):A problem of stochatic impulse control with discretionary stopping,In proceeding of the 39th IEEE Conference on Decision and Control,IEEE Control Systems Society,222-227,Piscataway,NJ.
[19]Duckworth,K.and M.Zervos(2001):A model for investment decisions with switching costs,Ann.Appl.Probab.11,239-260.
[20]Duffie D.and L.Epstein(1992):Stochastic differential utility,Econometrica 60(2),353-394.
[21]El Karoui,N.,C.Kapoudjian,E.Pardoux,S.Peng and M.C.Quenez(1997):Reflected solutions of backward SDEs and related obstacle problems for PDEs,Annals of Probability 25(2),702-737.
[22]F(o|¨)llmer,H.and A.Schied(2002):Convex measures of risk and trading constraints,Finance and Stochastics 6(4),429-447.
[23]F(o|¨)llmer,H.and A.Schied(2002):Robust preferences and convex measures of risk,in Sandmann,K.,Sch(o|¨)nbucher,P.J.(Eds.),Advances in Finance and Stochastics,39-56,Springer-Verlag.
[24]Frittelli,M.and E.Rosazza Gianin(2002):Putting order in risk measures,Journal of Banking and Finance 26(7),1473-1486.
[25]Gegout-Petit,A.and E.Pardoux(1996):Equations diff(?)rentielles stochastiques r(?)trogrades r(?)fl(?)chies dans un convexe,Stochastic Rep.57,111-128.
[26]Hamad(?)ne,S.,J.-P.Lepeltier and A.Matoussi(1997):Double barrier backward SDEs with continuous coefficient,In El Karoui,N.and L.Mazliak(Eds) Backward stochastic differential equations,Pitman Research Notes in Mathematics Series 364,161-177.
[27]Hamad(?)ne,S.(2002):Reflected BSDE's with discontinuous barrier and application,Stochastics Stochastic Rep.74(3-4),571-596.
[28]Hamad(?)ne,S.and M.Hassani(2005):BSDEs with two reflecting barriers:the general result,Probab.Theory Relat.Fields 132,237-264.
[29]Hamad(?)ne,S.and J.Zhang(2007):The Starting and Stopping Problem under Knightian Uncertainty and Related Systems of Reflected BSDEs.arXiv:0710.0908[math.PR]4 Oct 2007.
[30]Hu,Y.and S.Peng(2006):On the comparison theorem for multi-dimensional BSDEs,C.R.Math Acad.Sci.Paris 343,135-140.
[31]Hu,Y.and S.Tang(2007):Multi-dimensional BSDE with oblique reflection and optimal switching,Probab.Theory Related Fields,to appear,arXiv:0706.4365v2[math.PR]4 Jul 2007.
[32]Kallenberg,O.(2001):Foundations of modern probability,Springer.
[33]Karatzas,I.and S.E.Shreve(1988):Brownian Motion and Stochastic calculus,Springer.
[34]Karatzas,I.and S.E.Shreve(1998):Methods of Mathematical Finance,Springer.
[35]Kobylanski,M.(2000):Backward stochastic differential equations and partial differential equations with quadratic growth,Annals of Probability 28(2),558-602.
[36]Lazark,A.and M.C.Quenez(2003):A generalized stochastic differential utility,Mathematics of Operations Research 28(1),154-180.
[37]Lepeltier,J.-P.and M.Xu(2005):Penalization method for reflected backward stochastic differential equations with one r.c.l.l,barrier,Statistics Probability Letters 75,58-66.
[38]Li,J.and S.Tang(2007):A local strict comparison theorem and converse comparison theorems for reflected backward stochastic differential equations,Stochastic Process and Their Applications 117(9),1234-1250.
[39]L(u|¨)thi,H.-J.and J.Doege(2005):Convex risk measures for portfolio optimization and concepts of flexibility,Math.Program.,Ser.B 104,541-559.
[40]Meyer,P.A.(1976):Un cours sur les int(?)grales stochastiques,S(?)minaire de Probabilit(?)s X,Lecture Notes in.Math.511,245-400.Springer,Berlin.
[41]Pardoux,E.and S.Peng(1990):Adapted solution of a backward stochastic differential equation,Systems Control Lett.14,55-61.
[42]Peng,S.and M.Xu(2005):The smallest g-supermartingale and reflected BSDE with single and double L~2 obstacles,Ann.I.H.Poincare.PR 41,605-630.
[43]Rockafellar,R.T.and S.Uryasev(2000):Optimization of conditional value-at-risk,Journal of Risk 2(3),21-41.
[44]Schwartz,E.S.and L.Trigeorgis(2001):Real options and investment under uncertainty:classical readings and recent contributions,MIT press,Cambridge,MA.
[45]Tang,S.and S.Hou(2007):Switching games of stochastic differential systems,SIAM J.Control Optim.46(3),900-929.
[46]Trigeorgis,L.(1993):Real options and interactions with financial flexibility,Financial Management 22,202-224.
[47]Trigeorgis,L.(1996):Real option:Managerial flexibility and strategy in resource allocation,MIT press,Cambridge,MA.
[48]Yong,J.(1990):A zero-sum differential game in a finite duration with switching strategies,SIAM J.Control Optim.28(5),1234-1250.
[49]Yong,J.and X.Y.Zhou(1999):Stochastic controls:Hamiltonian Systems and HJB Equations,Springer-Verlag.
[50]Yong,J.(2004):Some estimates on exponentials of solutions to stochastic differential equations,Journal of applied mathematics and stochastic analysis 4,287-316.
[51]Zervos,M.(2003):A problem of sequential entry and exit decisions combined with discretionary stopping,SIAM J.Control Optim.42(2),397-421.
[52]雍炯敏(1992):动态规划方法与HAMILTON JACOBI-BELLMAN方程,上海科学技术出版社.
[2]Artzner,P.,F.Delbaen and J.M.Eber,D.Heath(1999):Coherent measures of risk,Mathematical Finance 9(3),203-228.
[3]Barrieu,P.and N.El Karoui(2004):Optimal Derivatives Design under Dynamic Risk Measures,Contemporary Mathematics,A.M.S.Proceedings.
[4]Basak,S.and A.Shapiro(2001):Value-at-Risk-Based Risk Management:Optimal Policies and Asset Prices,The review of financial studies summer 14(2),371-405.
[5]Benati,S.(2003):The optimal portfolio problem with coherent risk measure constraints,European Journal of Operational Research 150,572-584.
[6]Bensoussan,A.and J.L.Lions(1984):Impulse control and quasivariational inequalities,Ganthier-Villars,Montrouge.
[7]Brennan,M.J.and E.S.Schwartz(1985):Evaluating natural resource investments,J.Business 58,135-157.
[8]Briand,P.and Y.Shapiro(2006):BSDE with quadratic growth and unbounded terminal value,Probab.Theory Relat.Fields 136,604-618.
[9]Carmona,R.and M.Ludkovski(2005):Optimal Switching with Applications to Energy Tolling Agreements,preprint.
[10]Chen Z.and L.Epstein(2002):Ambiguity,risk and asset returns in continuous time Econometrica 70(4),1403-1443.
[11]Cvitanic,J.and I.Karatzas(1996):Backward SDEs with reflection and Dynkin games,Annals of Probability 24(4),2024-2056
[12]Delbaen,F.(2002):Coherent risk measures on general probability spaces,in Sandmann,K.(ed.) et al.,Advances in Finance and Stochastics,1-37,Springer-Verlag.
[13]Dellacherie,C.and P.A.Meyer(1980):Probabilit(?)s et Potentiel,Ⅰ-Ⅳ.Hermann,Paris.
[14]Detlefsen,K.and G.Scandolo(2005):Conditional and dynamic convex risk measures,Finance and Stochastics 9(4),539-561.
[15]Dixit,A.(1989):Entry and exit decisions under uncertainty,J.Political Economy 97,620-638.
[16]Dixit,A.and R.S.Pindyck(1994):Investment under uncertainty,Princeton Univ.Press.Princeton,NJ.
[17]Djehiche,B.and S.Hamad(?)ne(2007):On a finite horizon starting and stopping problem with default risk,preprint.
[18]Duckworth,K.and M.Zervos(2000):A problem of stochatic impulse control with discretionary stopping,In proceeding of the 39th IEEE Conference on Decision and Control,IEEE Control Systems Society,222-227,Piscataway,NJ.
[19]Duckworth,K.and M.Zervos(2001):A model for investment decisions with switching costs,Ann.Appl.Probab.11,239-260.
[20]Duffie D.and L.Epstein(1992):Stochastic differential utility,Econometrica 60(2),353-394.
[21]El Karoui,N.,C.Kapoudjian,E.Pardoux,S.Peng and M.C.Quenez(1997):Reflected solutions of backward SDEs and related obstacle problems for PDEs,Annals of Probability 25(2),702-737.
[22]F(o|¨)llmer,H.and A.Schied(2002):Convex measures of risk and trading constraints,Finance and Stochastics 6(4),429-447.
[23]F(o|¨)llmer,H.and A.Schied(2002):Robust preferences and convex measures of risk,in Sandmann,K.,Sch(o|¨)nbucher,P.J.(Eds.),Advances in Finance and Stochastics,39-56,Springer-Verlag.
[24]Frittelli,M.and E.Rosazza Gianin(2002):Putting order in risk measures,Journal of Banking and Finance 26(7),1473-1486.
[25]Gegout-Petit,A.and E.Pardoux(1996):Equations diff(?)rentielles stochastiques r(?)trogrades r(?)fl(?)chies dans un convexe,Stochastic Rep.57,111-128.
[26]Hamad(?)ne,S.,J.-P.Lepeltier and A.Matoussi(1997):Double barrier backward SDEs with continuous coefficient,In El Karoui,N.and L.Mazliak(Eds) Backward stochastic differential equations,Pitman Research Notes in Mathematics Series 364,161-177.
[27]Hamad(?)ne,S.(2002):Reflected BSDE's with discontinuous barrier and application,Stochastics Stochastic Rep.74(3-4),571-596.
[28]Hamad(?)ne,S.and M.Hassani(2005):BSDEs with two reflecting barriers:the general result,Probab.Theory Relat.Fields 132,237-264.
[29]Hamad(?)ne,S.and J.Zhang(2007):The Starting and Stopping Problem under Knightian Uncertainty and Related Systems of Reflected BSDEs.arXiv:0710.0908[math.PR]4 Oct 2007.
[30]Hu,Y.and S.Peng(2006):On the comparison theorem for multi-dimensional BSDEs,C.R.Math Acad.Sci.Paris 343,135-140.
[31]Hu,Y.and S.Tang(2007):Multi-dimensional BSDE with oblique reflection and optimal switching,Probab.Theory Related Fields,to appear,arXiv:0706.4365v2[math.PR]4 Jul 2007.
[32]Kallenberg,O.(2001):Foundations of modern probability,Springer.
[33]Karatzas,I.and S.E.Shreve(1988):Brownian Motion and Stochastic calculus,Springer.
[34]Karatzas,I.and S.E.Shreve(1998):Methods of Mathematical Finance,Springer.
[35]Kobylanski,M.(2000):Backward stochastic differential equations and partial differential equations with quadratic growth,Annals of Probability 28(2),558-602.
[36]Lazark,A.and M.C.Quenez(2003):A generalized stochastic differential utility,Mathematics of Operations Research 28(1),154-180.
[37]Lepeltier,J.-P.and M.Xu(2005):Penalization method for reflected backward stochastic differential equations with one r.c.l.l,barrier,Statistics Probability Letters 75,58-66.
[38]Li,J.and S.Tang(2007):A local strict comparison theorem and converse comparison theorems for reflected backward stochastic differential equations,Stochastic Process and Their Applications 117(9),1234-1250.
[39]L(u|¨)thi,H.-J.and J.Doege(2005):Convex risk measures for portfolio optimization and concepts of flexibility,Math.Program.,Ser.B 104,541-559.
[40]Meyer,P.A.(1976):Un cours sur les int(?)grales stochastiques,S(?)minaire de Probabilit(?)s X,Lecture Notes in.Math.511,245-400.Springer,Berlin.
[41]Pardoux,E.and S.Peng(1990):Adapted solution of a backward stochastic differential equation,Systems Control Lett.14,55-61.
[42]Peng,S.and M.Xu(2005):The smallest g-supermartingale and reflected BSDE with single and double L~2 obstacles,Ann.I.H.Poincare.PR 41,605-630.
[43]Rockafellar,R.T.and S.Uryasev(2000):Optimization of conditional value-at-risk,Journal of Risk 2(3),21-41.
[44]Schwartz,E.S.and L.Trigeorgis(2001):Real options and investment under uncertainty:classical readings and recent contributions,MIT press,Cambridge,MA.
[45]Tang,S.and S.Hou(2007):Switching games of stochastic differential systems,SIAM J.Control Optim.46(3),900-929.
[46]Trigeorgis,L.(1993):Real options and interactions with financial flexibility,Financial Management 22,202-224.
[47]Trigeorgis,L.(1996):Real option:Managerial flexibility and strategy in resource allocation,MIT press,Cambridge,MA.
[48]Yong,J.(1990):A zero-sum differential game in a finite duration with switching strategies,SIAM J.Control Optim.28(5),1234-1250.
[49]Yong,J.and X.Y.Zhou(1999):Stochastic controls:Hamiltonian Systems and HJB Equations,Springer-Verlag.
[50]Yong,J.(2004):Some estimates on exponentials of solutions to stochastic differential equations,Journal of applied mathematics and stochastic analysis 4,287-316.
[51]Zervos,M.(2003):A problem of sequential entry and exit decisions combined with discretionary stopping,SIAM J.Control Optim.42(2),397-421.
[52]雍炯敏(1992):动态规划方法与HAMILTON JACOBI-BELLMAN方程,上海科学技术出版社.