超前BSDE及超前BSDEwMS中的相关结果
|
摘要
倒向随机微分方程(简记为BSDE)的线性形式首先由Bismut在1973年[10]引入,1990年Pardoux和Peng在[5]中研究了Lipschitz条件下非线性BSDE解的存在唯一性定理.Duffe和Epstein在1992年[3]在研究随机微分效用过程中也独立地引进了一类倒向随机微分方程.在过去的近二十年中,BSDE倍受关注是因为它在随机控制、偏微分方程、数理金融、经济等领域都有着广泛的应用.
BSDE和非线性偏微方程之间存在着密切的联系(见Barles和Lesigne [7], Briand[2], Pardoux [4], Peng[6]等),并可应用于非线性半群理论及随机控制问题(见Quenez [16], El Karoui, Peng和Quenez [12], Hamadene和Lepeltier[24]及Peng[19]等),同时在金融数学中,套期保值及不确定权益定价理论通常可表为一类线性的BSDE(见El Karoui, Peng和Quenez [12]).在1997年Peng[20]引入了一种新的非线性期望：由特殊的BSDE导出的g-期望.利用Peng的g-期望很容易定义对应的条件期望.Rosazza[8]通过g-期望得到一类动态风险测度.Peng[18]定义了信息流相容的估计和g-估计,从而证明了满足某些特定限制的信息流相容的估计就是g-估计,即无论模型或机构如何来估值的,一旦它满足这些限制,在它背后就存在一个BSDE,生成元g是规则,而BSDE的解即是该估计.
另外,对两个随机微分方程比较定理的研究引起了相当多的关注(见Anderson [27], Ikeda和Watanable [17], Mao [28], Skorohod[1], Yamada[25],Yan[29]及Peng和Zhu[23]).基于上面的结果,Yang[22]研究了一类新的BSDE,即超前BSDE.发现超前BSDE和SDDE之间存在着完美的对偶,并给出了超前BSDE解的存在唯一性定理、比较定理、带停时的超前BSDE适应解的存在唯一性定理及带马尔可夫转换的随机微分延迟方程(SDDEwMS)的比较定理(见Chenggui Yuan, Zhe Yang和Xuerong Mao [15]).本文中研究了具有马尔可夫转换的BSDE和超前BSDE解的存在性和比较定理,超前BSDE解的稳定性定理并给出了具有连续系数的超前BSDE和非Lipschitz:条件的超前BSDE适应解的存在性唯一性定理.
文章组织如下：
第一章引入了基本的概念和定理.
第二章主要证明了超前BSDE解的稳定性,以及在连续系数和非Lipschitz条件下适应解的存在唯一性,以下是本章的主要结果：
定理2.1：(解的稳定性)假设超前BSDE满足(H3),(H4)和(H5),我们有
定理2.2：(适应解的存在性)(H6)线性增长条件(H7)对于固定的连续且递增.那么,如果下面的超前BSDE有适应解,即：存在Ft-适应的过程满足方程(2.3).
定理2.3：记(—Yi,—Zi),i=1,2,是下面方程的最小解,进一步,如果是方程(2.6)的最大解,那么定理2.4：(适应解的存在唯一性)设f满足条件(H2)和(H8),δ,ζ满足(1)和(2),则给定任意终端条件超前BSDE(1.3)有唯一解.
第三章主要讨论了具有马尔可夫转换的BSDE解的存在唯一性和比较定理,以下是本章的主要结果：
定理3.1：(适应解的存在唯一性)设f满足(H9)和(H10),那么对于任意的终端条件方程(3.1)有唯一解,也就是说：存在唯一一对Ft-适应过程满足方程(3.1).
定理3.2：(比较定理)设满足(H9)和(H10),j=1,2,令(Y.(j),Z.(j))分别表示下面BSDEwMS的解：且有严格比较定理：在上面的假设下,
在第四章我们研究了具有马尔可夫转换的超前BSDE解的存在唯一性和比较定理,以下是本章的主要结果：
定理4.1：(适应解的存在唯一性)设,满足(H1)’和(H2)’,δ,ζ满足(1)和(2).那么对于任给的终端条件,方程(4.1)有唯一解.
定理4.2：(比较定理)设fj满足(H1)’和(H2)’,
SF2(T,T+K),δ满足(1)和(2),且对于任意的是递增的,即：如果分别表示下面方程
的解：
且有严格比较定理：在上面的假设下,
The linear Backward Stochastic Differential Equation (BSDE for short) was first introduced by Bismut (1973) [10]. Pardoux和Peng (1990) [5] proved the existence and uniqueness theorem of the solution of nonlinear BSDE under Lipschitz condition. Duffe and Epstein (1992) [3] also proposed a type of BSDE independently to character-ize the stochastic differential utility. The theory of BSDEs has been studied with great interest in the last less than twenty years because of its connections with stochastic control, partial equation (PDE), mathematical finance and economics.
BSDE has great connections with nonlinear partial differential equations (see Bar-les and Lesigne [7], Briand [2], Pardoux [4], Peng [6], etc.) and can be used in non-linear semi-groups, and stochastic control problems (see Quenez [16], El Karoui,Peng and Quenez [12], Hamadene and Lepeltier [24] and Peng [19]). At the same time, in mathematical finance, the theory of the hedging and pricing of a contingent claim is typically expressed in terms of a linear BSDE (see El Karoui, Peng and Quenez [12]). In 1997 Peng [20] introduced a kind of nonlinear expectation:g-expectation via a par-ticular BSDE. Using Peng's g-expectation, it is easy to define conditional expectation. Rosazza [8] considered a type of dynamic risk measures via g-expectations. Peng [18] defined filtration consistent evaluation satisfying some restrictions is a g-evaluation, that is, whatever model or mechanism used to evaluate, once it satisfies the restrictions, there is a BSDE behind of it, the generator g is its mechanism, and the solution of BSDE is the evaluation.
On the other hand, the comparison theorems of two stochastic differential equa-tions have received a lot of attention (see Anderson [27], Ikeda and Watanable [17], Mao [28], Skorohod [1], Yamada [25], Yan [29], Peng and Zhu [23]). Based on the above results, Yang [22] researched a new type of equations:anticipated backward stochastic differential equations (anticipated BSDEs for short). There exists perfect du-ality between them and stochastic differential delay equations (SDDEs for short), and proved the existence and uniqueness theorem of the solutions of anticipated BSDEs and anticipated BSDEs with stopping times, the comparison theorem of the solutions of anticipated BSDEs and SDDEwMS (see Chenggui Yuan,Zhe Yang and Xuerong Mao [15]). In this paper, we prove the existence and uniqueness theorem and compar-ison theorem of the solutions of BSDEwMS and anticipated BSDEwMS. At the same time, we prove the existence and uniqueness theorem of anticipated BSDEs with con-tinuous coefficient and non-Lipschitz conditions, and prove the stability of the solutions of anticipated BSDEs.
The paper is organized as follows:Chapter 1 mainly introduces the basic concepts and lemmas. Chapter 2 proves the stability of the solutions of anticipated BSDEs, and the existence and uniqueness theorem of anticipated BSDEs with continuous coefficient and non-Lipschitz conditions. The following are the main results of Chapter 2.
Theorem 2.1:(Stability Theorem) If anticipated BSDE satisfies (H3), (H4) and (H5), we have
Theorem 2.2:(Existence Theorem of Adapted Solution) if f satisfies: (H6) linear growth: (H7) for fixed s,ω, y, z, f (s,ω,·,·,·) is continuous and f(s,ω,y,z,·) is increas-ing. Then, if the following anticipated BSDE has adapted solution, i.e., (?)-adapted process satisfies (2.3).
Theorem 2.3:Denote(-Yi,-Zi), i= 1,2, are the minimal solutions of the following equation, where fi satisfies (H1) and(H2), for further, if are the maximal solutions of (2.6), then
Theorem 2.4:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H2) and (H8),δ,ζsatisfy (1) and (2), then for any given terminal condition the anticipated BSDE (1.3) has a unique solution.
Chapter 3 studies the existence and uniqueness theorem and comparison theorem of the solutions of BSDEwMS. The following are the main results of Chapter 3.
Theorem 3.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H9) and (H10), then for any given terminal condition (3.1) has a unique solution, i.e., there exists a unique(?) -adapted process (Y., Z.)∈satisfies (3.1).
Theorem 3.2:(Comparison Theorem) If R satisfies (H9) and (H10), j= 1,2, let (Y(j)) and Z(j)) denote the solutions of the following BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,
Chapter 4 studies the existence and uniqueness theorem and comparison theorem of the solutions of anticipated BSDEwMS. The following are the main results of Chap-ter4.
Theorem 4.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H1)'and (H2)',δ,ζsatisfy (1) and (2). Then for any given terminal condition has a unique solution.
Theorem 4.2:(Comparison Theorem) If fj satisfies (H1)'and (H2)',j = 1,2,δsatisfies (1) and (2), and is increasing, i.e., if have denote the solutions of the fol-lowing anticipated BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,
BSDE和非线性偏微方程之间存在着密切的联系(见Barles和Lesigne [7], Briand[2], Pardoux [4], Peng[6]等),并可应用于非线性半群理论及随机控制问题(见Quenez [16], El Karoui, Peng和Quenez [12], Hamadene和Lepeltier[24]及Peng[19]等),同时在金融数学中,套期保值及不确定权益定价理论通常可表为一类线性的BSDE(见El Karoui, Peng和Quenez [12]).在1997年Peng[20]引入了一种新的非线性期望：由特殊的BSDE导出的g-期望.利用Peng的g-期望很容易定义对应的条件期望.Rosazza[8]通过g-期望得到一类动态风险测度.Peng[18]定义了信息流相容的估计和g-估计,从而证明了满足某些特定限制的信息流相容的估计就是g-估计,即无论模型或机构如何来估值的,一旦它满足这些限制,在它背后就存在一个BSDE,生成元g是规则,而BSDE的解即是该估计.
另外,对两个随机微分方程比较定理的研究引起了相当多的关注(见Anderson [27], Ikeda和Watanable [17], Mao [28], Skorohod[1], Yamada[25],Yan[29]及Peng和Zhu[23]).基于上面的结果,Yang[22]研究了一类新的BSDE,即超前BSDE.发现超前BSDE和SDDE之间存在着完美的对偶,并给出了超前BSDE解的存在唯一性定理、比较定理、带停时的超前BSDE适应解的存在唯一性定理及带马尔可夫转换的随机微分延迟方程(SDDEwMS)的比较定理(见Chenggui Yuan, Zhe Yang和Xuerong Mao [15]).本文中研究了具有马尔可夫转换的BSDE和超前BSDE解的存在性和比较定理,超前BSDE解的稳定性定理并给出了具有连续系数的超前BSDE和非Lipschitz:条件的超前BSDE适应解的存在性唯一性定理.
文章组织如下：
第一章引入了基本的概念和定理.
第二章主要证明了超前BSDE解的稳定性,以及在连续系数和非Lipschitz条件下适应解的存在唯一性,以下是本章的主要结果：
定理2.1：(解的稳定性)假设超前BSDE满足(H3),(H4)和(H5),我们有
定理2.2：(适应解的存在性)(H6)线性增长条件(H7)对于固定的连续且递增.那么,如果下面的超前BSDE有适应解,即：存在Ft-适应的过程满足方程(2.3).
定理2.3：记(—Yi,—Zi),i=1,2,是下面方程的最小解,进一步,如果是方程(2.6)的最大解,那么定理2.4：(适应解的存在唯一性)设f满足条件(H2)和(H8),δ,ζ满足(1)和(2),则给定任意终端条件超前BSDE(1.3)有唯一解.
第三章主要讨论了具有马尔可夫转换的BSDE解的存在唯一性和比较定理,以下是本章的主要结果：
定理3.1：(适应解的存在唯一性)设f满足(H9)和(H10),那么对于任意的终端条件方程(3.1)有唯一解,也就是说：存在唯一一对Ft-适应过程满足方程(3.1).
定理3.2：(比较定理)设满足(H9)和(H10),j=1,2,令(Y.(j),Z.(j))分别表示下面BSDEwMS的解：且有严格比较定理：在上面的假设下,
在第四章我们研究了具有马尔可夫转换的超前BSDE解的存在唯一性和比较定理,以下是本章的主要结果：
定理4.1：(适应解的存在唯一性)设,满足(H1)’和(H2)’,δ,ζ满足(1)和(2).那么对于任给的终端条件,方程(4.1)有唯一解.
定理4.2：(比较定理)设fj满足(H1)’和(H2)’,
SF2(T,T+K),δ满足(1)和(2),且对于任意的是递增的,即：如果分别表示下面方程
的解：
且有严格比较定理：在上面的假设下,
The linear Backward Stochastic Differential Equation (BSDE for short) was first introduced by Bismut (1973) [10]. Pardoux和Peng (1990) [5] proved the existence and uniqueness theorem of the solution of nonlinear BSDE under Lipschitz condition. Duffe and Epstein (1992) [3] also proposed a type of BSDE independently to character-ize the stochastic differential utility. The theory of BSDEs has been studied with great interest in the last less than twenty years because of its connections with stochastic control, partial equation (PDE), mathematical finance and economics.
BSDE has great connections with nonlinear partial differential equations (see Bar-les and Lesigne [7], Briand [2], Pardoux [4], Peng [6], etc.) and can be used in non-linear semi-groups, and stochastic control problems (see Quenez [16], El Karoui,Peng and Quenez [12], Hamadene and Lepeltier [24] and Peng [19]). At the same time, in mathematical finance, the theory of the hedging and pricing of a contingent claim is typically expressed in terms of a linear BSDE (see El Karoui, Peng and Quenez [12]). In 1997 Peng [20] introduced a kind of nonlinear expectation:g-expectation via a par-ticular BSDE. Using Peng's g-expectation, it is easy to define conditional expectation. Rosazza [8] considered a type of dynamic risk measures via g-expectations. Peng [18] defined filtration consistent evaluation satisfying some restrictions is a g-evaluation, that is, whatever model or mechanism used to evaluate, once it satisfies the restrictions, there is a BSDE behind of it, the generator g is its mechanism, and the solution of BSDE is the evaluation.
On the other hand, the comparison theorems of two stochastic differential equa-tions have received a lot of attention (see Anderson [27], Ikeda and Watanable [17], Mao [28], Skorohod [1], Yamada [25], Yan [29], Peng and Zhu [23]). Based on the above results, Yang [22] researched a new type of equations:anticipated backward stochastic differential equations (anticipated BSDEs for short). There exists perfect du-ality between them and stochastic differential delay equations (SDDEs for short), and proved the existence and uniqueness theorem of the solutions of anticipated BSDEs and anticipated BSDEs with stopping times, the comparison theorem of the solutions of anticipated BSDEs and SDDEwMS (see Chenggui Yuan,Zhe Yang and Xuerong Mao [15]). In this paper, we prove the existence and uniqueness theorem and compar-ison theorem of the solutions of BSDEwMS and anticipated BSDEwMS. At the same time, we prove the existence and uniqueness theorem of anticipated BSDEs with con-tinuous coefficient and non-Lipschitz conditions, and prove the stability of the solutions of anticipated BSDEs.
The paper is organized as follows:Chapter 1 mainly introduces the basic concepts and lemmas. Chapter 2 proves the stability of the solutions of anticipated BSDEs, and the existence and uniqueness theorem of anticipated BSDEs with continuous coefficient and non-Lipschitz conditions. The following are the main results of Chapter 2.
Theorem 2.1:(Stability Theorem) If anticipated BSDE satisfies (H3), (H4) and (H5), we have
Theorem 2.2:(Existence Theorem of Adapted Solution) if f satisfies: (H6) linear growth: (H7) for fixed s,ω, y, z, f (s,ω,·,·,·) is continuous and f(s,ω,y,z,·) is increas-ing. Then, if the following anticipated BSDE has adapted solution, i.e., (?)-adapted process satisfies (2.3).
Theorem 2.3:Denote(-Yi,-Zi), i= 1,2, are the minimal solutions of the following equation, where fi satisfies (H1) and(H2), for further, if are the maximal solutions of (2.6), then
Theorem 2.4:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H2) and (H8),δ,ζsatisfy (1) and (2), then for any given terminal condition the anticipated BSDE (1.3) has a unique solution.
Chapter 3 studies the existence and uniqueness theorem and comparison theorem of the solutions of BSDEwMS. The following are the main results of Chapter 3.
Theorem 3.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H9) and (H10), then for any given terminal condition (3.1) has a unique solution, i.e., there exists a unique(?) -adapted process (Y., Z.)∈satisfies (3.1).
Theorem 3.2:(Comparison Theorem) If R satisfies (H9) and (H10), j= 1,2, let (Y(j)) and Z(j)) denote the solutions of the following BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,
Chapter 4 studies the existence and uniqueness theorem and comparison theorem of the solutions of anticipated BSDEwMS. The following are the main results of Chap-ter4.
Theorem 4.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H1)'and (H2)',δ,ζsatisfy (1) and (2). Then for any given terminal condition has a unique solution.
Theorem 4.2:(Comparison Theorem) If fj satisfies (H1)'and (H2)',j = 1,2,δsatisfies (1) and (2), and is increasing, i.e., if have denote the solutions of the fol-lowing anticipated BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,
引文
[1]A.V.Skorokhod. Studies in the theory of random process. Addison-Wesley,Reading,MA,1965.
[2]Philippe Briand. Bsde's and viscosity solutions of semilinear pde's. Stochastics Stochastics Rep.64, (1-2):1-32,1998.
[3]D.Duffie and L.G.Epstein. Stochastic differential utility. Econometrica,60:353-394,1992.
[4]E.Pardoux and A.Yu.Veretennikov. Averaging of backward stochastic differential equations, with application to semi-linear pde's. Stochastics Stochastics Rep.60, (3-4):255-270,1997.
[5]E.Pardoux and S.Peng. Adapted solution of a backward stochastic differential equation. Sys-tems and Control Letters,14:55-61,1990.
[6]E.Pardoux and S.Peng. Backward stochastic differential equations and quasilinear parabolic partial differential equations, stochastic partial differential equations and their applica-tions(charlotte, nc,1991). Lecture Notes in Control and Inform.Sci.,176:200-217,1992.
[7]G.Barles and E.Lesigne. Sde.bsde and pde, backward stochastic differential equation. El Karoui and L.Mazliak, editors, Pitman Research Notes in Mat.Series,364,Longman,1997.
[8]E.Rosazza Gianin. Some examples of risk measures via g-expectations. working paper, ver-sion of 2004 May.
[9]Ying Hu and Shige Peng. A stability theorem of backward stochastic differential equations and its application. Probability Theory, pages 1059-1064,1997.
[10]J.M.Bismut. Theorie probabilitiste du controle des diffusion. Mem.Amer.Math.Soc, (176), 1973.
[11]J.P.Lepeltier and J.San Martion. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters,32:425-430,1997.
[12]El Karoui, S.Peng and M.C.Quenez. Backward stochastic differential equations in finance. Mathematical Finance,7(1):1-71,1997.
[13]Jicheng Liu and Jiagang Ren. Comparison theorem for solutions of backward stochastic dif-ferential equations with continuous coefficient. Statistics and Probability Letters,56:93-100, 2002.
[14]Xuerong Mao. Adapted solutions of backward stochastic differential equations with non-lipschitz coefficients. Stochastic Processes and their Applications,58:281-292,1995.
[15]Zhe Yang, Xuerong Mao and Chenggui Yuan. Comparison theorem of one-dimensional stochastic hybrid delay systems. Systems and Control Letters,57:56-63,2008.
[16]M.C.Quenez. MEthodes de controle stochastique en finance. These de doctorat de l' Universite Pierre et Marie Curie,1993.
[17]N.Ikeda and S.Watanable. A comparison theorem for solutions of stochastic differential equa-tions and its applications. Osaka J.Math.14, pages 619-633,1977.
[18]Shige Peng. Dynamical evaluations. C.R.Acad.Sci.Paris,339:585-589.
[19]Shige Peng. Backward stochastic differential equations and it's application in optimal control. Appl.Math.Optim.,27:125-144,1993.
[20]Shige Peng. Bsde and related g-expectation. In Pitman Research Notes in Mathematics Series, No.364, Backward Stochastic Differential Equation, ed. by N.El Karoui and L.Mazliak, pages 141-159,1997.
[21]Shige Peng. Nolinear expectations.nolinear evaluations and risk measures. Springer-Verlag Berlin Heidelberg,2004:165-253,2004.
[22]Shige Peng and Zhe Yang. Anticipated backward stochastic differential equations. The Annals of Probability,37(3):877-902,2009.
[23]Shige Peng and Xuehong Zhu. Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stochastic Processes and their Spplications, 116(3):370-380,2006.
[24]S.Hamadene and J.P.Lepeltier. Zero-sum stochastic differential games and backward equa-tions. Systems and Control Letters,24:259-263,1994.
[25]T.Yamada. On comparison theorem for solutions of stochastic differential equations and its applications. J.Math.Kyoto Univ.13, pages 497-512,1973.
[26]Ying Wang and Zhen Huang. Backward stochastic differential equations with non-lipschitz coefficients. Statistics and Probability Letters,79:1438-1443,2009.
[27]W.J.Anderson. Local behaviour of solutions of stochastic differential equations. Trans.Amer.Math.Soc.,164:309-321,1972.
[28]X.Mao. A note on comparison theorems for stochastic differential equations with respect to semimartingales. Stochastics Stochastics Rep.37, pages 49-59,1991.
[29]Jiaan Yan. A comparison theorem for semimartingales and its applications. Seminaire de Prob-abilites,XX,Lecture Notes in Mathematics,vol.1204,Springer,Berlin, pages 349-351,1986.
[2]Philippe Briand. Bsde's and viscosity solutions of semilinear pde's. Stochastics Stochastics Rep.64, (1-2):1-32,1998.
[3]D.Duffie and L.G.Epstein. Stochastic differential utility. Econometrica,60:353-394,1992.
[4]E.Pardoux and A.Yu.Veretennikov. Averaging of backward stochastic differential equations, with application to semi-linear pde's. Stochastics Stochastics Rep.60, (3-4):255-270,1997.
[5]E.Pardoux and S.Peng. Adapted solution of a backward stochastic differential equation. Sys-tems and Control Letters,14:55-61,1990.
[6]E.Pardoux and S.Peng. Backward stochastic differential equations and quasilinear parabolic partial differential equations, stochastic partial differential equations and their applica-tions(charlotte, nc,1991). Lecture Notes in Control and Inform.Sci.,176:200-217,1992.
[7]G.Barles and E.Lesigne. Sde.bsde and pde, backward stochastic differential equation. El Karoui and L.Mazliak, editors, Pitman Research Notes in Mat.Series,364,Longman,1997.
[8]E.Rosazza Gianin. Some examples of risk measures via g-expectations. working paper, ver-sion of 2004 May.
[9]Ying Hu and Shige Peng. A stability theorem of backward stochastic differential equations and its application. Probability Theory, pages 1059-1064,1997.
[10]J.M.Bismut. Theorie probabilitiste du controle des diffusion. Mem.Amer.Math.Soc, (176), 1973.
[11]J.P.Lepeltier and J.San Martion. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters,32:425-430,1997.
[12]El Karoui, S.Peng and M.C.Quenez. Backward stochastic differential equations in finance. Mathematical Finance,7(1):1-71,1997.
[13]Jicheng Liu and Jiagang Ren. Comparison theorem for solutions of backward stochastic dif-ferential equations with continuous coefficient. Statistics and Probability Letters,56:93-100, 2002.
[14]Xuerong Mao. Adapted solutions of backward stochastic differential equations with non-lipschitz coefficients. Stochastic Processes and their Applications,58:281-292,1995.
[15]Zhe Yang, Xuerong Mao and Chenggui Yuan. Comparison theorem of one-dimensional stochastic hybrid delay systems. Systems and Control Letters,57:56-63,2008.
[16]M.C.Quenez. MEthodes de controle stochastique en finance. These de doctorat de l' Universite Pierre et Marie Curie,1993.
[17]N.Ikeda and S.Watanable. A comparison theorem for solutions of stochastic differential equa-tions and its applications. Osaka J.Math.14, pages 619-633,1977.
[18]Shige Peng. Dynamical evaluations. C.R.Acad.Sci.Paris,339:585-589.
[19]Shige Peng. Backward stochastic differential equations and it's application in optimal control. Appl.Math.Optim.,27:125-144,1993.
[20]Shige Peng. Bsde and related g-expectation. In Pitman Research Notes in Mathematics Series, No.364, Backward Stochastic Differential Equation, ed. by N.El Karoui and L.Mazliak, pages 141-159,1997.
[21]Shige Peng. Nolinear expectations.nolinear evaluations and risk measures. Springer-Verlag Berlin Heidelberg,2004:165-253,2004.
[22]Shige Peng and Zhe Yang. Anticipated backward stochastic differential equations. The Annals of Probability,37(3):877-902,2009.
[23]Shige Peng and Xuehong Zhu. Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stochastic Processes and their Spplications, 116(3):370-380,2006.
[24]S.Hamadene and J.P.Lepeltier. Zero-sum stochastic differential games and backward equa-tions. Systems and Control Letters,24:259-263,1994.
[25]T.Yamada. On comparison theorem for solutions of stochastic differential equations and its applications. J.Math.Kyoto Univ.13, pages 497-512,1973.
[26]Ying Wang and Zhen Huang. Backward stochastic differential equations with non-lipschitz coefficients. Statistics and Probability Letters,79:1438-1443,2009.
[27]W.J.Anderson. Local behaviour of solutions of stochastic differential equations. Trans.Amer.Math.Soc.,164:309-321,1972.
[28]X.Mao. A note on comparison theorems for stochastic differential equations with respect to semimartingales. Stochastics Stochastics Rep.37, pages 49-59,1991.
[29]Jiaan Yan. A comparison theorem for semimartingales and its applications. Seminaire de Prob-abilites,XX,Lecture Notes in Mathematics,vol.1204,Springer,Berlin, pages 349-351,1986.