由连续鞅驱动的耦合的正倒向随机微分方程
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摘要
线性倒向随机微分方程是由Bismut于1978年首次提出的。Pardoux和Peng在1990年获得了非线性倒向随机微分方程在Lipschitz条件下解的存在唯一性定理。随后,许多学者进一步研究了倒向随机微分方程及其在数理金融,随机控制,偏微分方程,随机微分对策和经济等领域的应用。现在倒向随机微分方程理论不仅被广泛地认为是研究金融数学(例如,期权和衍生定价证券问题)的重要工具,而且也是研究随机控制,随机对策和非线性偏微分方程解的概率表示问题等的有效工具。经典的倒向随机微分方程理论是以Brown运动为干扰源的,而Brown运动是一种非常理想化的随机模型,致使经典的倒向随机微分方程的应用上受到很大的限制。
本文对经典的正倒向随机微分方程进行实质性的推广,将其干扰源由Brown运动推广为连续鞅,讨论了由连续鞅驱动的耦合的正倒向随机微分方程的若干问题。
由Brown运动驱动的耦合的正倒向随机微分方程首先是由Antonelli研究的。自从S.Peng和Zhen.Wu提出了完全耦合的的正倒向随机微分方程解的存在唯一性问题后,许多学者在此方面做了不懈的努力。在前人研究的基础上,在第三章中给出了了以连续鞅为干扰源的耦合的的正倒向随机微分方程解的存在唯一性条件,并利用纯概率方法给出了它的证明。研究这些问题时仍面临许多难题,例如,一般的鞅不再具有鞅表示定理的性质,也就是说,无法保证倒向随机微分方程解的存在性。对于这个问题,我们将通过假设鞅具有可料表示性来获得它。
Feynman-kac公式是随机分析中的一个很重要的公式。Peng利用经典的倒向随机微分方程对一大类二阶拟线性抛物型偏微分方程系统提供了一个概率解释,这一结果将著名的Feynman-kac公式推广到了非线性情形,即非线性Feynman-kac公式,它给出了常见的倒向随机微分方程的解与非线性偏微分方程的解的对应关系。在本文中,利用Ocone鞅的特殊性质,讨论了由Ocone鞅驱动的耦合的正倒向随机微分方程并把它和一类耦合了一个代数方程的拟线性抛物型偏微分方程系统联系起来,同时还给出了Feynman-kac公式的一个推广。
The linear backward stochastic differential equation was first proposed by Bismut in 1978.Then Pardoux and Peng first solved the existence and uniqueness theorem of the solution of the nonlinear BSDE under Lipschitz condition in 1990.From then on,many people make further study on BSDE and its applications in mathematical finance, stochastic control,partial differential equation(PDE),stochastic differential games and economy,which develop BSDE further.Now the theory of BSDE is not only widely considered as the main tool of study financial mathematial(for example,the problem of pricing of options and derivative securities) but also the efficient tool of studying stochastic control,stochastic games,the problem of probabilistic represention of solution of nonlinear PDE and so on.The classical BSDE is driven by Brownian Motion, but Brownian Motion is an ideal stochastic model which ristricts the applications of the classical BSDE.
In this paper,we generalize the classical BSDE essentially,its noise source by the Brownian Motion for continuous martingale,discussed some issues of Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale.
Coupled Forward-Backward Stochastic Differential Equations with Brownian Motion was first proposed by Antonelli.Since S.Peng and Zhen.Wu proposed the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations,many scholars have made the unrenmitting effort in this aspect.This paper based on the predecessor's work,first gives the condition of the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations driven by tne continuous martingale,and proves it by using purely probabilistic method.Considering these questions are still faced with many challenges,such as,general martingale is not Martingale said Theorem nature,in other words,there can be no assure BSDE the existence of solutions.On this issue,we will assume that Martingale has said can be expected to be its nature.
Feynman-kac formula is a important formula of Stochastic Analysis.Peng provides largest category of a second-order linear parabolic partial differential system with a probabilistic interpretation by using classic BSDE.The results promote famous Feynman-kac formula extended to the nonlinear case,Nonlinear Feynman-kac that formula,It is a common BSDE and the solution of nonlinear partial differential equations of the correspondence between solutions.In this paper,using the special nature of Ocone martingale,discussed by Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale and contact it with a kind of one-algebra The quasi-linear equations of parabolic partial differential equations systems to link up also given a formula Feynman-kac promotion.
本文对经典的正倒向随机微分方程进行实质性的推广,将其干扰源由Brown运动推广为连续鞅,讨论了由连续鞅驱动的耦合的正倒向随机微分方程的若干问题。
由Brown运动驱动的耦合的正倒向随机微分方程首先是由Antonelli研究的。自从S.Peng和Zhen.Wu提出了完全耦合的的正倒向随机微分方程解的存在唯一性问题后,许多学者在此方面做了不懈的努力。在前人研究的基础上,在第三章中给出了了以连续鞅为干扰源的耦合的的正倒向随机微分方程解的存在唯一性条件,并利用纯概率方法给出了它的证明。研究这些问题时仍面临许多难题,例如,一般的鞅不再具有鞅表示定理的性质,也就是说,无法保证倒向随机微分方程解的存在性。对于这个问题,我们将通过假设鞅具有可料表示性来获得它。
Feynman-kac公式是随机分析中的一个很重要的公式。Peng利用经典的倒向随机微分方程对一大类二阶拟线性抛物型偏微分方程系统提供了一个概率解释,这一结果将著名的Feynman-kac公式推广到了非线性情形,即非线性Feynman-kac公式,它给出了常见的倒向随机微分方程的解与非线性偏微分方程的解的对应关系。在本文中,利用Ocone鞅的特殊性质,讨论了由Ocone鞅驱动的耦合的正倒向随机微分方程并把它和一类耦合了一个代数方程的拟线性抛物型偏微分方程系统联系起来,同时还给出了Feynman-kac公式的一个推广。
The linear backward stochastic differential equation was first proposed by Bismut in 1978.Then Pardoux and Peng first solved the existence and uniqueness theorem of the solution of the nonlinear BSDE under Lipschitz condition in 1990.From then on,many people make further study on BSDE and its applications in mathematical finance, stochastic control,partial differential equation(PDE),stochastic differential games and economy,which develop BSDE further.Now the theory of BSDE is not only widely considered as the main tool of study financial mathematial(for example,the problem of pricing of options and derivative securities) but also the efficient tool of studying stochastic control,stochastic games,the problem of probabilistic represention of solution of nonlinear PDE and so on.The classical BSDE is driven by Brownian Motion, but Brownian Motion is an ideal stochastic model which ristricts the applications of the classical BSDE.
In this paper,we generalize the classical BSDE essentially,its noise source by the Brownian Motion for continuous martingale,discussed some issues of Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale.
Coupled Forward-Backward Stochastic Differential Equations with Brownian Motion was first proposed by Antonelli.Since S.Peng and Zhen.Wu proposed the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations,many scholars have made the unrenmitting effort in this aspect.This paper based on the predecessor's work,first gives the condition of the existence and uniqueness theorem of the solution of Coupled Forward-Backward Stochastic Differential Equations driven by tne continuous martingale,and proves it by using purely probabilistic method.Considering these questions are still faced with many challenges,such as,general martingale is not Martingale said Theorem nature,in other words,there can be no assure BSDE the existence of solutions.On this issue,we will assume that Martingale has said can be expected to be its nature.
Feynman-kac formula is a important formula of Stochastic Analysis.Peng provides largest category of a second-order linear parabolic partial differential system with a probabilistic interpretation by using classic BSDE.The results promote famous Feynman-kac formula extended to the nonlinear case,Nonlinear Feynman-kac that formula,It is a common BSDE and the solution of nonlinear partial differential equations of the correspondence between solutions.In this paper,using the special nature of Ocone martingale,discussed by Coupled Forward-Backward Stochastic Differential Equations driven by the continuous martingale and contact it with a kind of one-algebra The quasi-linear equations of parabolic partial differential equations systems to link up also given a formula Feynman-kac promotion.
引文
[1]Bismut J M.An introductory approach to duality in optimal stochastic control [J].SIAM Rev,1978,20(1):62-78
[2]Antonelli.Forward-backward stochastic differential equations[J].The Annals of Applied Probability,1993,3(3):777-793
[3]彭实戈.倒向随机微分方程及其应用.数学进展,1997
[4]黄志远.随机分析学基础(第二版).北京:科学出版社,2001
[5]Pardoux E,Peng S.Adapted solution of a backward stochastic differential equation.Systems Control Lett.,1990,14:55-61
[6]Cvitanic J,Ma J.Hedging options for a large investor ang forward-backward stochastic differential equations[J].Annals of Applied Probability,1996,6(2):370-389
[7]严加安,彭实戈,方涛赞,吴黎明.随机分析选讲.北京:科学出版社,2000
[8]Bismut J M.Conjugate Convex funtions in optimal control[J].Math.Anal.Appl.,1973,44:384-404
[9]Ma J,Protter P,Yong J.Solving forward-backward stochastic differential equations explicitly-a four step scheme[J].Probab.Theory and Related Fields,1994,98(2):339-359
[10]Pardoux E,Tang S.Forward-backward stochastic differential equations and quasilinear parabolic PDEs[J].Probab.Theory and Related Fields,1999,114(2):123-150
[11]Peng S,Wu Z.Fully coupled forward-backward stochastic differential equations and applications to optimal control[J].SIAM J.Control Optim.,1999,37(3):825-843
[12]Hu Y,Peng S.Solution of forward-backward stochastic differential equations[J].Probab.Theory and Related Fields,1995,103(2):273-283
[13]Yong J.Finding adapted solution of forward-backward stochastic differential equationsmethod of continuation[J].Probab.Theory and Related Fields,1997,107(4):537-572
[14]Yu Kabanov.On the Pontryagin maximum principle for the linear stochastic differential equations,Probabilistic Models and Control of Economical Processes.CEMI,Moscow,1978(in Russian)
[15]V Arkim,Yu Kabanov.Necessary optimality contions for stochastic differential equations.SovietMath.Dokl.,1979,20:1-5
[16]D Duffie,L Epstein.Asset pricing with stochatic differential utility.Rev.financial stud,1992b;5:411-436
[17]EI Karoui N,Quenez M C.Dynamic programming and pricing of contingent claims in an incomplete market.SIA M J.control.optim.,1995,33(1)
[18]Karatzas,I.Optimization problems in the theory of continunous trading.SIAM J.Control.,1984,27(6):1221-1259
[19]El Karoui,Peng S,Quenez M.Backward stochastic differential equation and finance,prepublication.1994,260:dulab.de probabilité
[20]El Karoui N,L Mazliak,eds.Backward stochastic differential equation.Longman,Harlow,1997
[21]S Hamadene,J-P Lepehier,Peng.BSDEs with continuous coefficients and stachistic differential games in Backward stochastic differential equations.N.El Karoui et al.,des.,Longman,Harlow,1997,115-128
[22]Tang S,Li X.Necessary contion for optimal control of stachitic systems with random jumps[J].SIAM J Control Optim,1994,32:1447-1457Pitman Research Notes in mathematics Series,1997:27-36
[24]Peng,S.Open problems on backward stochastic differential equations[A].eds.Boston:Kluwer Acad.Pub.,1999,265-273
[25]Ma J,Yong J.Solving forward-backward stochastic differential equations and the nodal set of Hamilton-Jacobi-Belhnan equations.Chin Ann.Math,ser.B,1995,16:279-298
[26]Hamadene S.Backward-forward stochastic differential equation's and stochastic differential games[J].Stochastic process and their Application,1998,77:1-15
[27]David Nualart,Wim Schoutens.Backward stochastic differential equation and Feyman-Kac formula for Levy processes with applications in finance.Bernoulli,2001,7(5):761-776
[28]Wang X J.On Backward stochastic differential equation by a continuous semimartingale.J.of Math.,1999,19
[29]XIA J.Backward stochastic differential equation with random measures[J].Acta mathematicae applicatae sinica,2000,16:3
[30]林清泉.倒向随机微分方程若干新结果.华中科技大学图书馆,1998
[31]龚光鲁.随机微分方程引论(第二版).北京:北京大学出版社,2000
[32]李娟.由一般鞅驱动的倒向随机微分方程.山东大学学报,2005
[33]Pardoux E,Peng S.Backward doubly stochastic differential equations and systems of quasi-linear parabolic SPDEs.Probab.Theory.Relat.Fields.,1994,98:209-227
[34]严加安.鞅与随机积分引论[M].上海:上海科学技术出版社,19981
[35]Peng S.Probalistic irterpretation for system of quasiiline parabolic partial differential equations.Stochastics,1991,37:61-74
[36]Peng S.A generalized dynamic programming principle and Hamilton-Jacob-Bellman equation.Stochastics.Rep,1992,38(121);119-134
[37]Pardoux E,Peng S.BSDE and quasilinear PPDE,SPDE and their applications (Charlotte,Nc,1991,)(B.L.Rozovskii and R.B.Sowers,eds),Lecture Notes in Control and Infbrm.Sci.,vol.176.Springer.Berlin,1992,200-217
[38]J Yong,X Zhou.Stochastic controls:Hamilton system and HJB equations.Springer New York,1999
[39]Peng S.A generalized Hamilton-Jacobi-Bellman equation.Lecture Notes in CIS 184,Li,Yong,eds.Spring-Verleg,Berlin,1991,126-134
[40]Hu Y.φksendal B.Fractional white noise calculus and applications to finance.Department of Mathematics.University of Oslo,1999,10,Preprint
[41]Lin S.Stochastic analysis of fractional Brownian motions.Stochastic and Stochastic Rep.,1995,55:121-140
[42]Shisekawa I.Sobolev spaces of Banach-valued functions associated with a Markov process.Probab.Theory Relat.Fields,1994,99(3):425-441
[43]Erraoui M,Nualart D,Ouknine Y.Hyperbolic stochastic partial diferential equations with additive fractional Brownian sheet.Stochastic Dynamics,2003,3:121-139
[44]李娟。由一般鞅驱动的倒向随机微分方程及应用.山东大学博士论文,2003
[45]何声武,汪嘉冈,严加安,半鞅与随机分析,北京:科学出版社,1995
[46]Bihari I.A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations.Acta Math.Acad.Sci.Hungar,1956,71-94
[47]Hu Y,Peng S.Adapeted solution of a backward semilinear stochastic evolution equation.Stochastic Anal.Appl.,1991,9:445-459
[2]Antonelli.Forward-backward stochastic differential equations[J].The Annals of Applied Probability,1993,3(3):777-793
[3]彭实戈.倒向随机微分方程及其应用.数学进展,1997
[4]黄志远.随机分析学基础(第二版).北京:科学出版社,2001
[5]Pardoux E,Peng S.Adapted solution of a backward stochastic differential equation.Systems Control Lett.,1990,14:55-61
[6]Cvitanic J,Ma J.Hedging options for a large investor ang forward-backward stochastic differential equations[J].Annals of Applied Probability,1996,6(2):370-389
[7]严加安,彭实戈,方涛赞,吴黎明.随机分析选讲.北京:科学出版社,2000
[8]Bismut J M.Conjugate Convex funtions in optimal control[J].Math.Anal.Appl.,1973,44:384-404
[9]Ma J,Protter P,Yong J.Solving forward-backward stochastic differential equations explicitly-a four step scheme[J].Probab.Theory and Related Fields,1994,98(2):339-359
[10]Pardoux E,Tang S.Forward-backward stochastic differential equations and quasilinear parabolic PDEs[J].Probab.Theory and Related Fields,1999,114(2):123-150
[11]Peng S,Wu Z.Fully coupled forward-backward stochastic differential equations and applications to optimal control[J].SIAM J.Control Optim.,1999,37(3):825-843
[12]Hu Y,Peng S.Solution of forward-backward stochastic differential equations[J].Probab.Theory and Related Fields,1995,103(2):273-283
[13]Yong J.Finding adapted solution of forward-backward stochastic differential equationsmethod of continuation[J].Probab.Theory and Related Fields,1997,107(4):537-572
[14]Yu Kabanov.On the Pontryagin maximum principle for the linear stochastic differential equations,Probabilistic Models and Control of Economical Processes.CEMI,Moscow,1978(in Russian)
[15]V Arkim,Yu Kabanov.Necessary optimality contions for stochastic differential equations.SovietMath.Dokl.,1979,20:1-5
[16]D Duffie,L Epstein.Asset pricing with stochatic differential utility.Rev.financial stud,1992b;5:411-436
[17]EI Karoui N,Quenez M C.Dynamic programming and pricing of contingent claims in an incomplete market.SIA M J.control.optim.,1995,33(1)
[18]Karatzas,I.Optimization problems in the theory of continunous trading.SIAM J.Control.,1984,27(6):1221-1259
[19]El Karoui,Peng S,Quenez M.Backward stochastic differential equation and finance,prepublication.1994,260:dulab.de probabilité
[20]El Karoui N,L Mazliak,eds.Backward stochastic differential equation.Longman,Harlow,1997
[21]S Hamadene,J-P Lepehier,Peng.BSDEs with continuous coefficients and stachistic differential games in Backward stochastic differential equations.N.El Karoui et al.,des.,Longman,Harlow,1997,115-128
[22]Tang S,Li X.Necessary contion for optimal control of stachitic systems with random jumps[J].SIAM J Control Optim,1994,32:1447-1457Pitman Research Notes in mathematics Series,1997:27-36
[24]Peng,S.Open problems on backward stochastic differential equations[A].eds.Boston:Kluwer Acad.Pub.,1999,265-273
[25]Ma J,Yong J.Solving forward-backward stochastic differential equations and the nodal set of Hamilton-Jacobi-Belhnan equations.Chin Ann.Math,ser.B,1995,16:279-298
[26]Hamadene S.Backward-forward stochastic differential equation's and stochastic differential games[J].Stochastic process and their Application,1998,77:1-15
[27]David Nualart,Wim Schoutens.Backward stochastic differential equation and Feyman-Kac formula for Levy processes with applications in finance.Bernoulli,2001,7(5):761-776
[28]Wang X J.On Backward stochastic differential equation by a continuous semimartingale.J.of Math.,1999,19
[29]XIA J.Backward stochastic differential equation with random measures[J].Acta mathematicae applicatae sinica,2000,16:3
[30]林清泉.倒向随机微分方程若干新结果.华中科技大学图书馆,1998
[31]龚光鲁.随机微分方程引论(第二版).北京:北京大学出版社,2000
[32]李娟.由一般鞅驱动的倒向随机微分方程.山东大学学报,2005
[33]Pardoux E,Peng S.Backward doubly stochastic differential equations and systems of quasi-linear parabolic SPDEs.Probab.Theory.Relat.Fields.,1994,98:209-227
[34]严加安.鞅与随机积分引论[M].上海:上海科学技术出版社,19981
[35]Peng S.Probalistic irterpretation for system of quasiiline parabolic partial differential equations.Stochastics,1991,37:61-74
[36]Peng S.A generalized dynamic programming principle and Hamilton-Jacob-Bellman equation.Stochastics.Rep,1992,38(121);119-134
[37]Pardoux E,Peng S.BSDE and quasilinear PPDE,SPDE and their applications (Charlotte,Nc,1991,)(B.L.Rozovskii and R.B.Sowers,eds),Lecture Notes in Control and Infbrm.Sci.,vol.176.Springer.Berlin,1992,200-217
[38]J Yong,X Zhou.Stochastic controls:Hamilton system and HJB equations.Springer New York,1999
[39]Peng S.A generalized Hamilton-Jacobi-Bellman equation.Lecture Notes in CIS 184,Li,Yong,eds.Spring-Verleg,Berlin,1991,126-134
[40]Hu Y.φksendal B.Fractional white noise calculus and applications to finance.Department of Mathematics.University of Oslo,1999,10,Preprint
[41]Lin S.Stochastic analysis of fractional Brownian motions.Stochastic and Stochastic Rep.,1995,55:121-140
[42]Shisekawa I.Sobolev spaces of Banach-valued functions associated with a Markov process.Probab.Theory Relat.Fields,1994,99(3):425-441
[43]Erraoui M,Nualart D,Ouknine Y.Hyperbolic stochastic partial diferential equations with additive fractional Brownian sheet.Stochastic Dynamics,2003,3:121-139
[44]李娟。由一般鞅驱动的倒向随机微分方程及应用.山东大学博士论文,2003
[45]何声武,汪嘉冈,严加安,半鞅与随机分析,北京:科学出版社,1995
[46]Bihari I.A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations.Acta Math.Acad.Sci.Hungar,1956,71-94
[47]Hu Y,Peng S.Adapeted solution of a backward semilinear stochastic evolution equation.Stochastic Anal.Appl.,1991,9:445-459
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