带随机违约时间的倒向随机微分方程，超前倒向随机微分方程及其相关结果
|
摘要
形如的方程被称为倒向随机微分方程(简记为BSDE),其线性形式由Bismut [14]在1973年引入,其一般形式由Pardoux和Peng [73]于1990年首次研究.
在过去的近二十年中,BSDE理论受到了广泛的关注(参见[1],[2],[4],[20],[21],[22],[28],[54],[62],[63],[64],[68],[83],[90],[91],等等).特别地,比较定理是这一理论的一大重要结果,这归因于Peng [79],然后由Pardoux和Peng [74], El Karoui et al.[39],Hu和Peng [55]做了推广.比较定理告诉我们这样一个事实：当我们可以比较两个BSDE的终端条件和生成元时,那么我们也可以对其解做出比较.反过来,我们也有逆比较定理,即当可以比较解时那也可以比较生成元,参见Briand et al. [19], Coquetet a1.[30],Hu和Peng [55], Jiang [59]等.
在无违约市场中,这些结果被得到了广泛的应用.准确地讲,BSDE在金融数学中有广泛的应用,如在定价和对冲理论中的应用(参见El Karoui et al. [39]等),在随机控制和对策理论中的应用(参见Buckdahn和Li [23], El Karoui et al. [39], El Karoui和Hamadene [36], Hamadene [44], Hamadene和Lepeltier [45]及[46], Hamadene et al. [47], Hamadene et al. [48]及[49], Peng [80], Quenez [85], Peng和Wu[81],等等),在偏微分方程(简记为PDE)中的应用(参见Barles et al. [5], Barles和Lesigne [6], Briand [18], Pardoux和Peng [74], Pardoux和Tang [76], Pardoux和Veretennikov [77], Peng[78]及[79],Wu和Yu[88],等等).
同时,人们也对BSDE的反射解做了大量研究.单边界反射BSDE由El Karoui et al. [37]首次提出,其解被保持在一个给定的随机过程(称为边界或者障碍)的上方.双边界反射BSDE也随之被引入(见Cvitanic和Karatzas [32], Hamadene et al. [47]等).事实上反射BSDE是一个非常热门的课题,因为它在金融,对策,控制问题,偏微分方程等各方面都有重要的应用(参见Bally et al. [3], El Karoui et al. [38], Hamadene和Lepeltier [46], Lepeltier et al. [61], Matoussi [70])后来这些结果被推广到不连续障碍的情况(参见Hamadene [43], Lepeltier和Xu [65], Peng和Xu [82]). Hamadene和Ouknine [50](也可参见Hamadene和Wang [51])研究了由布朗运动和一个与布朗运动独立的泊松随机测度所驱动的反射BSDE.
本文的主要目的是丰富并改进BSDE理论,以下是本文的主要结果.
第一章：介绍本文工作的背景及第二章到第四章所研究的主要问题.
第二章：在违约框架下,我们首次引入了一种新型BSDE一一带随机违约时间的BSDE,它由布朗运动及一个与布朗运动独立的不连续鞅所驱动,其一般形式为：其中是一个与B独立的(?)-鞅,τ.是随机违约时间,(?)是扩充的信息流.
对于此类方程,我们有如下结果.值得一提的是,对生成元的要求,比较定理要强于存在唯一性定理.
定理2.2.2. (存在唯一性定理)假设g满足(a2.1)和(a2.2),那么对于任意给定的终端条件,上述BSDE存在唯一解
定理2.2.7.(比较定理)记(Y,Z,ζ),(Y,Z,ζ)分别为如下两个一维带随机违约时间的BSDE的唯一解:其中ξ,ξ满足的条件与定理2.2.2相同,g满足(a2.1)-(a2.3),gs∈Lg2(0,T；R).如果那么
此外,如下结果成立俨格比较定理):
然后我们处理了一种特殊情况——更一般地,通过带随机违约时间的BSDE,我们对违约风险的PDE途径做了介绍.对此,我们有
定理2.4.3.假设如下定义的函数u：那么u具有如下概率解释:并且我们有：其中由如下两个方程唯一确定:
作为应用,我们研究了违约框架下的零和随机微分对策问题.假设两个博弈者J1和J2在同一控制系统中进行博弈.受控系统的机制如下：对应于控制u∈u和v∈v的值函数(J1的代价,J2的酬劳)如下：条件期望值函数为.
为了解决此问题,我们定义与此对策问题相关的Hamilton函数：
假设Isaacs' condition成立.记(u*,υ*)(t,x,z,ζ)为函数H的鞍点,并记
主要结果为
定理2.5.3.带随机违约时间的BSDE存在唯一解此外,二元组(u*,v*)是对策问题的鞍点.
第三章：我们研究了如下一般形式的超前BSDE(简记为ABSDE):其中δ(·)：[0,T]→R+,ζ(·)：[0,T]→R+为满足一定条件的连续函数.
对于一类特殊的高维ABSDE(生成元不含Z的超前项且关于Y的超前项不一定递增),我们建立了如下比较定理：
定理3.3.5.下面两条是等价的:(i)对任意且ξ(1)≥ξ(2),如下ABSDE的唯一解满足(ii)对任意且θ(1)≥θ(2),其中c>0为一常数.
第四章：研究了如下一般形式的推广的ABSDE(简记为GABSDE):
对于一维GABSDE,我们给出如下一般的比较定理：
定理4.2.3.记分别为为如下GABSDE的唯一解(j=1,2):那么Yt(1)≥Yt(2),a.e,a.s..
此外,我们也讨论了带有连续障碍S的GABSDE.
定理4.3.3.假设(A4.1)-(A4.3)成立,那么反射GABSDE存在唯一解
非常幸运,利用上述结果我们可以研究一种特殊情形——带泛函障碍≥St的反射方程.对此,我们有
定理4.4.5.假设(a4.1)-(a4.3)成立,障碍S是如下形式的半鞅:其中分别为取值于R,Rd的循序可测过程,并满足那么反射GABSDE至少具有一个解并且二元组是唯一确定的.
Backward stochastic differential equations (BSDEs in short) of the following gen-eral form were introduced, in the linear case, by Bismut [14] in 1973 and considered the general form the first time by Pardoux-Peng [73] in 1990.
In the last twenty years, the theory of BSDEs has been studied with great interest (see e.g. [1], [2], [4], [20], [21], [22], [28], [54], [62], [63], [64], [68], [83], [90], [91], etc.). Particularly, the comparison theorem turns out to be one of the achievements of this theory. It is due to Peng [79] and then generalized by Pardoux-Peng [74], El Karoui et al. [39], Hu-Peng [55]. It allows to compare the solutions of two BSDEs whenever we can compare the terminal conditions and the generators. Conversely, we can also compare the generators if we can compare the solutions, see e.g. Briand et al. [19], Coquet et al. [30], Hu-Peng [55], Jiang [59].
These results are applied widely to default-free markets. Precisely, BSDE was applied widely in financial mathematics, such as the pricing/hedging problem (see e.g. El Karoui et al. [39], etc.), in the stochastic control and game theory (see e.g. Buckdahn-Li [23], El Karoui et al. [39], El Karoui-Hamadene [36], Hamadene [44], Hamadene-Lepeltier [45] and [46], Hamadene et al. [47], Hamadene et al. [48] and [49], Peng [80], Quenez [85], Peng-Wu [81], etc.), and in the theory of partial differential equations (PDEs in short) (see e.g. Barles et al. [5], Barles-Lesigne [6], Briand [18], Pardoux-Peng [74], Pardoux-Tang [76], Pardoux-Veretennikov [77], Peng [78] and [79], Wu-Yu [88], etc.).
In the meantime, people also have a good study of the reflected solutions to BS-DEs, that is, the solution is forced to stay above a given stochastic process which is called the obstacle. More precisely, reflected BSDE, with one barrier introduced by El Karoui et al. [37], with double barrier studied by, e.g., Cvitanic-Karatzas [32] and Hamadene et al. [47], is also a hot topic due to its wide applications to finance, the game or control problems and partial differential equations (see e.g. Bally et al. [3], El Karoui et al. [38], Hamadene-Lepeltier [46], Lepeltier et al. [61], Matoussi [70]). These results were then generalized to the case where the obstacle is discontinuous (see e.g. Hamadene [43], Lepeltier-Xu [65], Peng-Xu [82]), and then by Hamadene-Ouknine [50] (see also Hamadene-Wang [51]) to the discontinuous case where the reflected BSDE is driven by a Brownian motion and an independent Poisson random measure.
The objective of this thesis is to enrich and improve the theory of BSDEs. In the following, we list the main results of this thesis.
Chapter 1:In this chapter, we present the motivations of our work and list the main problems studied from Chapter 2 to Chapter 4.
Chapter 2:In this chapter, we introduce a new type of BSDE in a defaultable set-up, called BSDE with random default time, which is driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. These equations are of the following general form: whereγsds is a G-martingale independent of B,τis the random default time and G is the enlarged filtration.
For these equations, we have the following result. It should be mentioned here that the comparison theorem requires more conditions than the existence and uniqueness theorem.
Theorem 2.2.2. (Existence and Uniqueness Theorem) Assume that g satis-fies (a2.1) and (a2.2), then for any fixed terminal conditionξ∈L2(GT;Rm), the above BSDE has a unique solution
Theorem 2.2.7. (Comparison Theorem) Let (Y, Z,ζ), (Y, Z,ζ) be the unique solu-tions of the following two 1-dimensional BSDEs with random default time, respectively: whereξ,ξsatisfy the same assumptions as in Theorem 2.2.2, g satisfies (a2.1)-(a2.3), gs∈LG2(0,T;R).If then
Besides, the following holds true (the strict comparison theorem):
Then we deal with a particular case where More generally, we introduce the PDE approach to default risk via BSDE with random default time. For this, we have
Theorem 2.4.3. Assume that the function u, defined by satisfies, for any given (t, x)∈[0, T]×Rm,
Then u has the following probabilistic representation:
Moreover, the following hold: where (Yt,x, Zt,x,ζt,x) is determined uniquely by and
As one application, we deal with an application in zero-sum stochastic differential games in a defaultable setting. Assume that two players J1 and J2 intervene on a system with antagonistic advantages. The dynamics of the controlled system is The cost functional corresponding to u∈U andυ∈V is given by which is a cost (resp. reward) for J1 (resp. J2).The conditional expected remaining cost from time
In order to tackle this problem, we define the Hamilton function associated with this game problem as following: Assume that Isaacs'condition is fulfilled. Denote by (u*,υ*)(t,x,z,ζ) the saddle point for the function H. Write
The main result is
Theorem 2.5.3. The BSDE with random default time has a unique solution which satisfies where Moreover, the pair (u*,v*) is a saddle point for the game.
Chapter 3:In this chapter, we consider the anticipated BSDEs (ABSDEs) of the following form: whereδ(·):[0,T]→R+ andζ(·):[0, T]→R+ are continuous functions satisfying certain conditions.
We establish the following comparison theorem for multidimensional ABSDEs with generators independent of the anticipated term of Z and possibly not increasing in the anticipated term of Y:
Theorem 3.3.5. The following are equivalent: (i) for all the unique solutions to the following ABSDE: satisfy where c> 0 is a constant.
Chapter 4:In this chapter, we study the generalized ABSDE (GABSDE) of the following form:
For the 1-dimensional GABSDEs, we give a more general comparison theorem as follows:
Theorem 4.2.3. Let (j= 1,2) be the unique solutions to GABSDEs respectively: where j= 1,2, fj satisfies C;Rd). If uous semimartingale and then Yt(1)≥Yt(2),a.e.,a.s..
Moreover, we are also concerned with the real-valued reflected GABSDE with one continuous barrier S.
Theorem 4.3.3. Assume that (A4.1)-(A4.3) hold, then the reflected GABSDE has a unique solution
Fortunately, the above result can be applied to deal with a particular case when the obstacle is of functional form For this, we have Theorem 4.4.5. Assume that (a4.1)-(a4.3) hold and moreover that the obstacle S is a semimartingale of the form where (μt)t∈[0,T] and (σt)t∈[0,T] are progressively measurable processes, with values in R and Rd respectively, satisfying
Then the reflected GABSDE has at least a solution and the pair is uniquely determined.
在过去的近二十年中,BSDE理论受到了广泛的关注(参见[1],[2],[4],[20],[21],[22],[28],[54],[62],[63],[64],[68],[83],[90],[91],等等).特别地,比较定理是这一理论的一大重要结果,这归因于Peng [79],然后由Pardoux和Peng [74], El Karoui et al.[39],Hu和Peng [55]做了推广.比较定理告诉我们这样一个事实：当我们可以比较两个BSDE的终端条件和生成元时,那么我们也可以对其解做出比较.反过来,我们也有逆比较定理,即当可以比较解时那也可以比较生成元,参见Briand et al. [19], Coquetet a1.[30],Hu和Peng [55], Jiang [59]等.
在无违约市场中,这些结果被得到了广泛的应用.准确地讲,BSDE在金融数学中有广泛的应用,如在定价和对冲理论中的应用(参见El Karoui et al. [39]等),在随机控制和对策理论中的应用(参见Buckdahn和Li [23], El Karoui et al. [39], El Karoui和Hamadene [36], Hamadene [44], Hamadene和Lepeltier [45]及[46], Hamadene et al. [47], Hamadene et al. [48]及[49], Peng [80], Quenez [85], Peng和Wu[81],等等),在偏微分方程(简记为PDE)中的应用(参见Barles et al. [5], Barles和Lesigne [6], Briand [18], Pardoux和Peng [74], Pardoux和Tang [76], Pardoux和Veretennikov [77], Peng[78]及[79],Wu和Yu[88],等等).
同时,人们也对BSDE的反射解做了大量研究.单边界反射BSDE由El Karoui et al. [37]首次提出,其解被保持在一个给定的随机过程(称为边界或者障碍)的上方.双边界反射BSDE也随之被引入(见Cvitanic和Karatzas [32], Hamadene et al. [47]等).事实上反射BSDE是一个非常热门的课题,因为它在金融,对策,控制问题,偏微分方程等各方面都有重要的应用(参见Bally et al. [3], El Karoui et al. [38], Hamadene和Lepeltier [46], Lepeltier et al. [61], Matoussi [70])后来这些结果被推广到不连续障碍的情况(参见Hamadene [43], Lepeltier和Xu [65], Peng和Xu [82]). Hamadene和Ouknine [50](也可参见Hamadene和Wang [51])研究了由布朗运动和一个与布朗运动独立的泊松随机测度所驱动的反射BSDE.
本文的主要目的是丰富并改进BSDE理论,以下是本文的主要结果.
第一章：介绍本文工作的背景及第二章到第四章所研究的主要问题.
第二章：在违约框架下,我们首次引入了一种新型BSDE一一带随机违约时间的BSDE,它由布朗运动及一个与布朗运动独立的不连续鞅所驱动,其一般形式为：其中是一个与B独立的(?)-鞅,τ.是随机违约时间,(?)是扩充的信息流.
对于此类方程,我们有如下结果.值得一提的是,对生成元的要求,比较定理要强于存在唯一性定理.
定理2.2.2. (存在唯一性定理)假设g满足(a2.1)和(a2.2),那么对于任意给定的终端条件,上述BSDE存在唯一解
定理2.2.7.(比较定理)记(Y,Z,ζ),(Y,Z,ζ)分别为如下两个一维带随机违约时间的BSDE的唯一解:其中ξ,ξ满足的条件与定理2.2.2相同,g满足(a2.1)-(a2.3),gs∈Lg2(0,T；R).如果那么
此外,如下结果成立俨格比较定理):
然后我们处理了一种特殊情况——更一般地,通过带随机违约时间的BSDE,我们对违约风险的PDE途径做了介绍.对此,我们有
定理2.4.3.假设如下定义的函数u：那么u具有如下概率解释:并且我们有：其中由如下两个方程唯一确定:
作为应用,我们研究了违约框架下的零和随机微分对策问题.假设两个博弈者J1和J2在同一控制系统中进行博弈.受控系统的机制如下：对应于控制u∈u和v∈v的值函数(J1的代价,J2的酬劳)如下：条件期望值函数为.
为了解决此问题,我们定义与此对策问题相关的Hamilton函数：
假设Isaacs' condition成立.记(u*,υ*)(t,x,z,ζ)为函数H的鞍点,并记
主要结果为
定理2.5.3.带随机违约时间的BSDE存在唯一解此外,二元组(u*,v*)是对策问题的鞍点.
第三章：我们研究了如下一般形式的超前BSDE(简记为ABSDE):其中δ(·)：[0,T]→R+,ζ(·)：[0,T]→R+为满足一定条件的连续函数.
对于一类特殊的高维ABSDE(生成元不含Z的超前项且关于Y的超前项不一定递增),我们建立了如下比较定理：
定理3.3.5.下面两条是等价的:(i)对任意且ξ(1)≥ξ(2),如下ABSDE的唯一解满足(ii)对任意且θ(1)≥θ(2),其中c>0为一常数.
第四章：研究了如下一般形式的推广的ABSDE(简记为GABSDE):
对于一维GABSDE,我们给出如下一般的比较定理：
定理4.2.3.记分别为为如下GABSDE的唯一解(j=1,2):那么Yt(1)≥Yt(2),a.e,a.s..
此外,我们也讨论了带有连续障碍S的GABSDE.
定理4.3.3.假设(A4.1)-(A4.3)成立,那么反射GABSDE存在唯一解
非常幸运,利用上述结果我们可以研究一种特殊情形——带泛函障碍≥St的反射方程.对此,我们有
定理4.4.5.假设(a4.1)-(a4.3)成立,障碍S是如下形式的半鞅:其中分别为取值于R,Rd的循序可测过程,并满足那么反射GABSDE至少具有一个解并且二元组是唯一确定的.
Backward stochastic differential equations (BSDEs in short) of the following gen-eral form were introduced, in the linear case, by Bismut [14] in 1973 and considered the general form the first time by Pardoux-Peng [73] in 1990.
In the last twenty years, the theory of BSDEs has been studied with great interest (see e.g. [1], [2], [4], [20], [21], [22], [28], [54], [62], [63], [64], [68], [83], [90], [91], etc.). Particularly, the comparison theorem turns out to be one of the achievements of this theory. It is due to Peng [79] and then generalized by Pardoux-Peng [74], El Karoui et al. [39], Hu-Peng [55]. It allows to compare the solutions of two BSDEs whenever we can compare the terminal conditions and the generators. Conversely, we can also compare the generators if we can compare the solutions, see e.g. Briand et al. [19], Coquet et al. [30], Hu-Peng [55], Jiang [59].
These results are applied widely to default-free markets. Precisely, BSDE was applied widely in financial mathematics, such as the pricing/hedging problem (see e.g. El Karoui et al. [39], etc.), in the stochastic control and game theory (see e.g. Buckdahn-Li [23], El Karoui et al. [39], El Karoui-Hamadene [36], Hamadene [44], Hamadene-Lepeltier [45] and [46], Hamadene et al. [47], Hamadene et al. [48] and [49], Peng [80], Quenez [85], Peng-Wu [81], etc.), and in the theory of partial differential equations (PDEs in short) (see e.g. Barles et al. [5], Barles-Lesigne [6], Briand [18], Pardoux-Peng [74], Pardoux-Tang [76], Pardoux-Veretennikov [77], Peng [78] and [79], Wu-Yu [88], etc.).
In the meantime, people also have a good study of the reflected solutions to BS-DEs, that is, the solution is forced to stay above a given stochastic process which is called the obstacle. More precisely, reflected BSDE, with one barrier introduced by El Karoui et al. [37], with double barrier studied by, e.g., Cvitanic-Karatzas [32] and Hamadene et al. [47], is also a hot topic due to its wide applications to finance, the game or control problems and partial differential equations (see e.g. Bally et al. [3], El Karoui et al. [38], Hamadene-Lepeltier [46], Lepeltier et al. [61], Matoussi [70]). These results were then generalized to the case where the obstacle is discontinuous (see e.g. Hamadene [43], Lepeltier-Xu [65], Peng-Xu [82]), and then by Hamadene-Ouknine [50] (see also Hamadene-Wang [51]) to the discontinuous case where the reflected BSDE is driven by a Brownian motion and an independent Poisson random measure.
The objective of this thesis is to enrich and improve the theory of BSDEs. In the following, we list the main results of this thesis.
Chapter 1:In this chapter, we present the motivations of our work and list the main problems studied from Chapter 2 to Chapter 4.
Chapter 2:In this chapter, we introduce a new type of BSDE in a defaultable set-up, called BSDE with random default time, which is driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. These equations are of the following general form: whereγsds is a G-martingale independent of B,τis the random default time and G is the enlarged filtration.
For these equations, we have the following result. It should be mentioned here that the comparison theorem requires more conditions than the existence and uniqueness theorem.
Theorem 2.2.2. (Existence and Uniqueness Theorem) Assume that g satis-fies (a2.1) and (a2.2), then for any fixed terminal conditionξ∈L2(GT;Rm), the above BSDE has a unique solution
Theorem 2.2.7. (Comparison Theorem) Let (Y, Z,ζ), (Y, Z,ζ) be the unique solu-tions of the following two 1-dimensional BSDEs with random default time, respectively: whereξ,ξsatisfy the same assumptions as in Theorem 2.2.2, g satisfies (a2.1)-(a2.3), gs∈LG2(0,T;R).If then
Besides, the following holds true (the strict comparison theorem):
Then we deal with a particular case where More generally, we introduce the PDE approach to default risk via BSDE with random default time. For this, we have
Theorem 2.4.3. Assume that the function u, defined by satisfies, for any given (t, x)∈[0, T]×Rm,
Then u has the following probabilistic representation:
Moreover, the following hold: where (Yt,x, Zt,x,ζt,x) is determined uniquely by and
As one application, we deal with an application in zero-sum stochastic differential games in a defaultable setting. Assume that two players J1 and J2 intervene on a system with antagonistic advantages. The dynamics of the controlled system is The cost functional corresponding to u∈U andυ∈V is given by which is a cost (resp. reward) for J1 (resp. J2).The conditional expected remaining cost from time
In order to tackle this problem, we define the Hamilton function associated with this game problem as following: Assume that Isaacs'condition is fulfilled. Denote by (u*,υ*)(t,x,z,ζ) the saddle point for the function H. Write
The main result is
Theorem 2.5.3. The BSDE with random default time has a unique solution which satisfies where Moreover, the pair (u*,v*) is a saddle point for the game.
Chapter 3:In this chapter, we consider the anticipated BSDEs (ABSDEs) of the following form: whereδ(·):[0,T]→R+ andζ(·):[0, T]→R+ are continuous functions satisfying certain conditions.
We establish the following comparison theorem for multidimensional ABSDEs with generators independent of the anticipated term of Z and possibly not increasing in the anticipated term of Y:
Theorem 3.3.5. The following are equivalent: (i) for all the unique solutions to the following ABSDE: satisfy where c> 0 is a constant.
Chapter 4:In this chapter, we study the generalized ABSDE (GABSDE) of the following form:
For the 1-dimensional GABSDEs, we give a more general comparison theorem as follows:
Theorem 4.2.3. Let (j= 1,2) be the unique solutions to GABSDEs respectively: where j= 1,2, fj satisfies C;Rd). If uous semimartingale and then Yt(1)≥Yt(2),a.e.,a.s..
Moreover, we are also concerned with the real-valued reflected GABSDE with one continuous barrier S.
Theorem 4.3.3. Assume that (A4.1)-(A4.3) hold, then the reflected GABSDE has a unique solution
Fortunately, the above result can be applied to deal with a particular case when the obstacle is of functional form For this, we have Theorem 4.4.5. Assume that (a4.1)-(a4.3) hold and moreover that the obstacle S is a semimartingale of the form where (μt)t∈[0,T] and (σt)t∈[0,T] are progressively measurable processes, with values in R and Rd respectively, satisfying
Then the reflected GABSDE has at least a solution and the pair is uniquely determined.
引文
[1]F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab.3 (1993) 777-793.
[2]F. Antonelli, S. Hamadene, Existence of the solutions of backward-forward SDEs with continuous monotone coefficients, Statistics and Probability Letter 76 (2006) 1559-1569.
[3]V. Bally, M. E. Caballero, B. Fernandez, N. El Karoui, Reflected BSDE's, PDE's and Variational Inequalities, Rapport de recherche INRIA No.4455 (2004).
[4]V. Bally, A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, Journal of Theoretical Probability 14 (2001) 125-164.
[5]G. Barles, R. Buckdahn, E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep.60 (1997) 57-83.
[6]G. Barles, E. Lesigne, SDE, BSDE and PDE, Backward stochastic differential equations, Pitman Res. Notes Math. Ser.364, Longman,1997, pp.47-80.
[7]V. E. Benes, Existence of optimal stochastic control laws, SIAM Journal on Con-trol Optimization 9(3) (1971) 446-472.
[8]T. R. Bielecki, S. Crepey, M. Jeanblanc, M. Rutkowski, Defaultable options in a markovian intensity model of credit risk, Mathematical Finance 18(4) (2008) 493-518.
[9]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Hedging of defaultable claims, Paris-Princeton Lecture on Mathematical Finance 2003, Lecture Notes in Mathematics 1847 (2004) 1-132. Springer.
[10]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Modelling and valuation of credit risk. In:M.Frittelli and W.Runggaldier (eds), CIME-EMS Summer School on Stochastic Methods in Finance, Bressanone, Lecture Notes in Mahtematics 1856 (2004) 27-126, Springer.
[11]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, PDE approach to valuation and hedging of credit derivatives, Quantitative Finance 5 (2005) 257-270.
[12]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Introduction to mathematics of credit risk modeling, Stochastic Models in Mathematical Finance, CIMPA-UNESCO-MOROCCO School, Marrakech, Morocco, April 9-20,2007, pp.1-78.
[13]T. R. Bielecki, M. Rutkowski, Credit Risk:Modelling, Valuation and Hedging, Springer,2002.
[14]J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Apl.44 (1973) 384-404.
[15]F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(3) (1973) 637-654.
[16]C. Bluhm, L. Overbeck, C. Wagner, An Introduction to Credit Risk Modelling, Chapman& Hall,2004.
[17]P. Bremaud, M. Yor, Changes of filtration and of probability measures, Z. Wahr. Verw. Gebiete 45 (1978) 269-295.
[18]P. Briand, BSDE's and viscosity solutions of semilinear PDE's, Stochastics An International Journal of Probability and Stochastic Processes 64 (1-2) (1998) 1-32.
[19]P. Briand, F. Coquet, Y. Hu, J. Memin, S. Peng, A converse comparison theorem for BSDEs and related properties of g-expectation, Electron Comm Probab 5 (2000) 101-117.
[20]P. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Stoica, Lp solutions of BSDEs, Stochastic Process. Appl.108 (2003) 109-129.
[21]P. Briand, Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields 136(4) (2006) 604-618.
[22]P. Briand, Y. Hu, Quadratic BSDEs with convex generators and unbounded ter-minal conditions, Probability Theory and Related Fields 141(4) (2008) 543-567.
[23]R. Buckdahn, J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM Journal of Control and Opti-mization 47 (2008) 444-475.
[24]R. Buckdahn, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations (Part Ⅰ), Stochastic Processes and their Applications 93 (2001) 181-204.
[25]R. Buckdahn, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations (Part Ⅱ), Stochastic Processes and their Applications 93 (2001) 205-228.
[26]R. Buckdahn, M. Quincampoix, A.Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab Theory Relat Fields 116 (2000) 485-504.
[27]Z. Chen, L. Epstein, Ambiguity, risk, and asset returns in continous time, Econo-metrica 70(4) (2002) 1403-1443.
[28]Z. Chen, R. Kulperger, G. Wei, A comonotonic theorem for BSDEs, Stochastic Processes and their Applications 115 (2005),41-54.
[29]Z. Chen, S. Peng, A general downcrossing inequality for g-martingales, Statistics and Probability Letters 46 (2000) 169-175.
[30]F. Coquet, Y. Hu, J. Memin, S. Peng, A general converse comparison theorem for backward stochastic differential equations, C R Acad Sci Paris, Ser I Math 333 (2001) 577-581.
[31]M. G. Crandall, H. Ishii, P. L. Lions, User's guide to viscosity solution of second order partial differential equation, Amer. Math. Soc.27(1) (1992) 1-67.
[32]J. Cvitanic, I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab.24(4) (1996) 2024-2056.
[33]C. Dellacherie, P. A. Meyer, Probabilities and Potential B:Theory of Martingales; North-Holland, Amsterdam,1982.
[34]C. Dellacherie, P. A. Meyer, Probabilites et Potentiel, Chap. I-IV, Paris:Her-mann,1975.
[35]C. Dellacherie, P. A. Meyer, Probabilites et Potentiel, Chap. V-VIII, Paris:Her-mann,1980.
[36]N. El Karoui, S. Hamadene, BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Processes and Their Applications 107 (2003) 145-169.
[37]N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M. C. Quenez, Reflected solution of backward SDE and related obstacle problems for PDEs, Ann. Probab. 25(2) (1997) 702-737.
[38]N. El Karoui, E. Pardoux, M. C. Quenez, Reflected backward SDEs and American options. In:L.C.G.Rogers and D.Talay (eds), Numerical Methods in Finance, Cambridge:Cambridge University Press (1997) 215-231.
[39]N. El Karoui, S. Peng, M. Quenez, Backward stochastic differential equations in finance, Mathematical Finance 7(1) (1997) 1-71.
[40]N. El Karoui, M. Quenez, Dynamic programming and pricing of contingent claims in incomplete market, Siam J. Control Optim.33 (1995) 29-66.
[41]H. Engler, S. M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-Jacobi Equations, Proc. London Math. Soc.63 (1991) 212-240.
[42]J. Gregory, J. P. Laurent, I will survive, Risk, June (2003) 103-107.
[43]S. Hamadene, Reflected BSDE's with discontinuous barrier and application, Stochastics and Stochastic Reports,74 (2002) 571-596.
[44]S. Hamadene, Mixed zero-sum stochastic differential game and american game options, SIAM J.Control Optim.45 (2006) 496-518.
[45]S. Hamadene, J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems and Control Letters 24 (1995) 259-263.
[46]S. Hamadene, J. P. Lepeltier, Reflected BSDEs and mixed game problems, Stochastic processes and their applications 85 (2000) 177-188.
[47]S. Hamadene, J. P. Lepeltier, A. Matoussi, Double barrier backward SDEs with continuous coefficient. In:N.El Karoui and L.Mazliak (eds), Pitman Research Notes in Mathematics Series 364 (1997) 161-177.
[48]S. Hamadene, J. P. Lepeltier, S. Peng, BSDEs with continuous coefficients and stochastic differential games. In:EL. Karoui and L. Mazliak (eds), Pitman Re-search Notes in Mathematics Series 364 (1997) 115-128, Langman, Harlow.
[49]S. Hamadene, J. P. Lepeltier, Z. WU, Infinite horizon reflected backward stochas-tic differential equation and applications in mixed control and game problems, Probability and Mathematical Statistics 19 (1999) 211-234.
[50]S. Hamadene, Y. Ouknine, Reflected Backward stochastic differential equation with jumps and random obstacle, EJP 8 (2003) 1-20.
[51]S. Hamadene, H. Wang, BSDEs with two rcll reflecting obstacles driven by a Brow-nian motion and Poisson measure and related mixed zero-sum games, Stochastic Processes and their Applications,2009. doi=10.1016/j.spa.2009.03.004.
[55]M. Harrison, S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stoch. Process. Appl.11 (1981) 215-260.
[53]S. He, J. Wang, J. Yan, Semimartingale Theory and Stochastic Calculus, Science Press and CRC Press,1992.
[54]Y. Hu, S. Peng, Solution of forward-backward stochastic differential equations, Prob. Theory and Rel. Fields 103 (1995) 273-283.
[55]Y. Hu, S. Peng, On the comparison theorem for multidimensional BSDEs, C R Acad Sci Paris, Ser.I 343 (2006) 135-140.
[56]H. Ishii, S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Communications in partial differential equations 16(6-7) (1991) 1095-1128.
[57]R. A. Jarrow, F. Yu, Counterparty risk and the pricing of defaultable securities, Journal of Finance 56 (2001) 1765-1799.
[58]M. Jeanblanc, M. Rutkowski, Modelling default risk:An overview. In:Mathemat-ical Finance:Theory and Practice, pp.171-269. High Education Press. Beijing, 2000.
[59]L. Jiang, Converse comparison theorems for backward stochastic differential equa-tions, statistics and probability letters 71 (2005) 173-183.
[60]S. Kusuoka, A remark on default risk models, Advances in Mathematical Eco-nomics 1 (1999) 69-82.
[61]J. P. Lepeltier, A. Matoussi, M. Xu, Reflected BSDEs under monotonicity and general increasing growth conditiond, Advanced in applied probability, March (2005) 1-26.
[62]J. P. Lepeltier, J. San Martin, Backward stochastic differential equations with continuous coefficients, Statistics and Proba. Letters 34 (1997) 425-430.
[63]J. P. Lepeltier, J. San Martin, Existence for BSDE with superlinear-quadratic coefficient, Stochastic and Stochastic Reports 63 (1998) 227-240.
[64]J. P. Lepeltier, J. San Martin, BSDE's with continuous, monotonicity, and non-Lipschitz in z coefficient,2004. Preprint.
[65]J. P. Lepeltier, M. Xu, Penalization method for reflected backward stochastic differential equations with one rcll barrier, Statistics and Probability Letters 75 (2004) 58-66.
[66]T. Lim, M. C. Quenez, Utility maximization in incomplete markets with default, pdf-file available in arXiv:0811.4715v2 [math.OC] 22 Mar 2009.
[67]W. Liu, Y. Yang, G. Lu, Viscosity solutions of fully nonlinear parabolic systems, J. Math. Anal. Appl.281 (2003) 362-381.
[68]J. Ma, P. Protter, J. Yong, Solving forward-backward stochastic differential equa-tions explicitly-a four step scheme, Probab. Theory and Related Fields 98 (1994) 339-359.
[69]R. Mansuy, M. Yor, Random times and enlagements of filtrations in a Brownian setting, Lecture Notes in Mathematics, vol.1873,2006, Springer.
[70]A. Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficient, Statistic and Probality Letters 34 (1997) 347-354.
[71]G. Mazziotto, J. Szpirglas, Modele General de filtrage non lineaire et equations differentielles stochastiques associees, Ann. Inst. Henri Poincare 15 (1979) 147-173.
[72]R. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci.4 (1973) 141-183.
[73]E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equa-tion, Systems and Control Letters 14 (1990) 55-61.
[74]E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci.176 (1992) 200-217, Springer, Berlin.
[75]E. Pardoux, F. Pradeilles, Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Annales de l'Institut Henri Poincare (B) Probability and Statistics 33(4) (1997) 467-490.
[76]E. Pardoux, S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields 114 (1999) 123-150.
[77]E. Pardoux, A. Yu. Veretennikov, Averaging of backward stochastic differential equations, with application to semi-linear PDE's, Stochastics and stochastics re-ports 60 (3-4) (1997) 255-270.
[78]S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports 37 (1991) 61-74.
[79]S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics and Stochastics Reports 38 (1992) 119-134.
[80]S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim.27 (1993) 125-144.
[81]S. Peng, Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control and Optim.37 (1999) 825-843.
[82]S. Peng, M. Xu, The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles, Annale de l'I. H. P., PR 41 (2005) 605-630.
[83]S. Peng, Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability 37(3) (2009) 877-902.
[84]P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, second edition,2004.
[85]M. C. Quenez, Methodes de controle stochastique en Finance, These de doctorat de l'Universite Pierre et Marie Curie,1993.
[86]D. Revuz, M. Yor, Continuous Matingales and Brownian Motion,3ed., Springer, New York,1999.
[87]Y. Saisho, Stochastic differential equations for multidimensional domains with reflecting boundary, Prob. Theory and Rel. Fields 74 (1987) 455-477.
[88]Z. Wu, Z. Yu, Fully coupled forward-backward stochastic differential equations and related partial differential equations system, Chinese Journal of Contempo-rary Mathematics 3(2004) 269-282.
[89]Z. Yang, Generalized backward stochastic differential equation. Journal of Shan-dong University (Natural Science) 41(6) (2006) 81-86.
[90]Z. Yang, Anticipated BSDEs and related results in SDEs, Shandong University Doctoral Dissertation, ID:237010200420228,2007, pp.1-72.
[91]J. Yong, Finding adapted solution of forward-backward stochastic differential equations method of continuation, Probab. Theory and Rel. Fields 107 (1997) 537-572.
[92]严加安,彭实戈,方诗赞,吴黎明：随机分析选讲,科学出版社,1997.
[93]严加安：鞅与随机积分引论,上海科学技术出版社,1981.
[94]严加安：测度论讲义,科学出版社,1998.
[2]F. Antonelli, S. Hamadene, Existence of the solutions of backward-forward SDEs with continuous monotone coefficients, Statistics and Probability Letter 76 (2006) 1559-1569.
[3]V. Bally, M. E. Caballero, B. Fernandez, N. El Karoui, Reflected BSDE's, PDE's and Variational Inequalities, Rapport de recherche INRIA No.4455 (2004).
[4]V. Bally, A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, Journal of Theoretical Probability 14 (2001) 125-164.
[5]G. Barles, R. Buckdahn, E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep.60 (1997) 57-83.
[6]G. Barles, E. Lesigne, SDE, BSDE and PDE, Backward stochastic differential equations, Pitman Res. Notes Math. Ser.364, Longman,1997, pp.47-80.
[7]V. E. Benes, Existence of optimal stochastic control laws, SIAM Journal on Con-trol Optimization 9(3) (1971) 446-472.
[8]T. R. Bielecki, S. Crepey, M. Jeanblanc, M. Rutkowski, Defaultable options in a markovian intensity model of credit risk, Mathematical Finance 18(4) (2008) 493-518.
[9]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Hedging of defaultable claims, Paris-Princeton Lecture on Mathematical Finance 2003, Lecture Notes in Mathematics 1847 (2004) 1-132. Springer.
[10]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Modelling and valuation of credit risk. In:M.Frittelli and W.Runggaldier (eds), CIME-EMS Summer School on Stochastic Methods in Finance, Bressanone, Lecture Notes in Mahtematics 1856 (2004) 27-126, Springer.
[11]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, PDE approach to valuation and hedging of credit derivatives, Quantitative Finance 5 (2005) 257-270.
[12]T. R. Bielecki, M. Jeanblanc, M. Rutkowski, Introduction to mathematics of credit risk modeling, Stochastic Models in Mathematical Finance, CIMPA-UNESCO-MOROCCO School, Marrakech, Morocco, April 9-20,2007, pp.1-78.
[13]T. R. Bielecki, M. Rutkowski, Credit Risk:Modelling, Valuation and Hedging, Springer,2002.
[14]J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Apl.44 (1973) 384-404.
[15]F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(3) (1973) 637-654.
[16]C. Bluhm, L. Overbeck, C. Wagner, An Introduction to Credit Risk Modelling, Chapman& Hall,2004.
[17]P. Bremaud, M. Yor, Changes of filtration and of probability measures, Z. Wahr. Verw. Gebiete 45 (1978) 269-295.
[18]P. Briand, BSDE's and viscosity solutions of semilinear PDE's, Stochastics An International Journal of Probability and Stochastic Processes 64 (1-2) (1998) 1-32.
[19]P. Briand, F. Coquet, Y. Hu, J. Memin, S. Peng, A converse comparison theorem for BSDEs and related properties of g-expectation, Electron Comm Probab 5 (2000) 101-117.
[20]P. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Stoica, Lp solutions of BSDEs, Stochastic Process. Appl.108 (2003) 109-129.
[21]P. Briand, Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields 136(4) (2006) 604-618.
[22]P. Briand, Y. Hu, Quadratic BSDEs with convex generators and unbounded ter-minal conditions, Probability Theory and Related Fields 141(4) (2008) 543-567.
[23]R. Buckdahn, J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM Journal of Control and Opti-mization 47 (2008) 444-475.
[24]R. Buckdahn, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations (Part Ⅰ), Stochastic Processes and their Applications 93 (2001) 181-204.
[25]R. Buckdahn, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations (Part Ⅱ), Stochastic Processes and their Applications 93 (2001) 205-228.
[26]R. Buckdahn, M. Quincampoix, A.Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab Theory Relat Fields 116 (2000) 485-504.
[27]Z. Chen, L. Epstein, Ambiguity, risk, and asset returns in continous time, Econo-metrica 70(4) (2002) 1403-1443.
[28]Z. Chen, R. Kulperger, G. Wei, A comonotonic theorem for BSDEs, Stochastic Processes and their Applications 115 (2005),41-54.
[29]Z. Chen, S. Peng, A general downcrossing inequality for g-martingales, Statistics and Probability Letters 46 (2000) 169-175.
[30]F. Coquet, Y. Hu, J. Memin, S. Peng, A general converse comparison theorem for backward stochastic differential equations, C R Acad Sci Paris, Ser I Math 333 (2001) 577-581.
[31]M. G. Crandall, H. Ishii, P. L. Lions, User's guide to viscosity solution of second order partial differential equation, Amer. Math. Soc.27(1) (1992) 1-67.
[32]J. Cvitanic, I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab.24(4) (1996) 2024-2056.
[33]C. Dellacherie, P. A. Meyer, Probabilities and Potential B:Theory of Martingales; North-Holland, Amsterdam,1982.
[34]C. Dellacherie, P. A. Meyer, Probabilites et Potentiel, Chap. I-IV, Paris:Her-mann,1975.
[35]C. Dellacherie, P. A. Meyer, Probabilites et Potentiel, Chap. V-VIII, Paris:Her-mann,1980.
[36]N. El Karoui, S. Hamadene, BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Processes and Their Applications 107 (2003) 145-169.
[37]N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M. C. Quenez, Reflected solution of backward SDE and related obstacle problems for PDEs, Ann. Probab. 25(2) (1997) 702-737.
[38]N. El Karoui, E. Pardoux, M. C. Quenez, Reflected backward SDEs and American options. In:L.C.G.Rogers and D.Talay (eds), Numerical Methods in Finance, Cambridge:Cambridge University Press (1997) 215-231.
[39]N. El Karoui, S. Peng, M. Quenez, Backward stochastic differential equations in finance, Mathematical Finance 7(1) (1997) 1-71.
[40]N. El Karoui, M. Quenez, Dynamic programming and pricing of contingent claims in incomplete market, Siam J. Control Optim.33 (1995) 29-66.
[41]H. Engler, S. M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-Jacobi Equations, Proc. London Math. Soc.63 (1991) 212-240.
[42]J. Gregory, J. P. Laurent, I will survive, Risk, June (2003) 103-107.
[43]S. Hamadene, Reflected BSDE's with discontinuous barrier and application, Stochastics and Stochastic Reports,74 (2002) 571-596.
[44]S. Hamadene, Mixed zero-sum stochastic differential game and american game options, SIAM J.Control Optim.45 (2006) 496-518.
[45]S. Hamadene, J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems and Control Letters 24 (1995) 259-263.
[46]S. Hamadene, J. P. Lepeltier, Reflected BSDEs and mixed game problems, Stochastic processes and their applications 85 (2000) 177-188.
[47]S. Hamadene, J. P. Lepeltier, A. Matoussi, Double barrier backward SDEs with continuous coefficient. In:N.El Karoui and L.Mazliak (eds), Pitman Research Notes in Mathematics Series 364 (1997) 161-177.
[48]S. Hamadene, J. P. Lepeltier, S. Peng, BSDEs with continuous coefficients and stochastic differential games. In:EL. Karoui and L. Mazliak (eds), Pitman Re-search Notes in Mathematics Series 364 (1997) 115-128, Langman, Harlow.
[49]S. Hamadene, J. P. Lepeltier, Z. WU, Infinite horizon reflected backward stochas-tic differential equation and applications in mixed control and game problems, Probability and Mathematical Statistics 19 (1999) 211-234.
[50]S. Hamadene, Y. Ouknine, Reflected Backward stochastic differential equation with jumps and random obstacle, EJP 8 (2003) 1-20.
[51]S. Hamadene, H. Wang, BSDEs with two rcll reflecting obstacles driven by a Brow-nian motion and Poisson measure and related mixed zero-sum games, Stochastic Processes and their Applications,2009. doi=10.1016/j.spa.2009.03.004.
[55]M. Harrison, S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stoch. Process. Appl.11 (1981) 215-260.
[53]S. He, J. Wang, J. Yan, Semimartingale Theory and Stochastic Calculus, Science Press and CRC Press,1992.
[54]Y. Hu, S. Peng, Solution of forward-backward stochastic differential equations, Prob. Theory and Rel. Fields 103 (1995) 273-283.
[55]Y. Hu, S. Peng, On the comparison theorem for multidimensional BSDEs, C R Acad Sci Paris, Ser.I 343 (2006) 135-140.
[56]H. Ishii, S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Communications in partial differential equations 16(6-7) (1991) 1095-1128.
[57]R. A. Jarrow, F. Yu, Counterparty risk and the pricing of defaultable securities, Journal of Finance 56 (2001) 1765-1799.
[58]M. Jeanblanc, M. Rutkowski, Modelling default risk:An overview. In:Mathemat-ical Finance:Theory and Practice, pp.171-269. High Education Press. Beijing, 2000.
[59]L. Jiang, Converse comparison theorems for backward stochastic differential equa-tions, statistics and probability letters 71 (2005) 173-183.
[60]S. Kusuoka, A remark on default risk models, Advances in Mathematical Eco-nomics 1 (1999) 69-82.
[61]J. P. Lepeltier, A. Matoussi, M. Xu, Reflected BSDEs under monotonicity and general increasing growth conditiond, Advanced in applied probability, March (2005) 1-26.
[62]J. P. Lepeltier, J. San Martin, Backward stochastic differential equations with continuous coefficients, Statistics and Proba. Letters 34 (1997) 425-430.
[63]J. P. Lepeltier, J. San Martin, Existence for BSDE with superlinear-quadratic coefficient, Stochastic and Stochastic Reports 63 (1998) 227-240.
[64]J. P. Lepeltier, J. San Martin, BSDE's with continuous, monotonicity, and non-Lipschitz in z coefficient,2004. Preprint.
[65]J. P. Lepeltier, M. Xu, Penalization method for reflected backward stochastic differential equations with one rcll barrier, Statistics and Probability Letters 75 (2004) 58-66.
[66]T. Lim, M. C. Quenez, Utility maximization in incomplete markets with default, pdf-file available in arXiv:0811.4715v2 [math.OC] 22 Mar 2009.
[67]W. Liu, Y. Yang, G. Lu, Viscosity solutions of fully nonlinear parabolic systems, J. Math. Anal. Appl.281 (2003) 362-381.
[68]J. Ma, P. Protter, J. Yong, Solving forward-backward stochastic differential equa-tions explicitly-a four step scheme, Probab. Theory and Related Fields 98 (1994) 339-359.
[69]R. Mansuy, M. Yor, Random times and enlagements of filtrations in a Brownian setting, Lecture Notes in Mathematics, vol.1873,2006, Springer.
[70]A. Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficient, Statistic and Probality Letters 34 (1997) 347-354.
[71]G. Mazziotto, J. Szpirglas, Modele General de filtrage non lineaire et equations differentielles stochastiques associees, Ann. Inst. Henri Poincare 15 (1979) 147-173.
[72]R. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci.4 (1973) 141-183.
[73]E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equa-tion, Systems and Control Letters 14 (1990) 55-61.
[74]E. Pardoux, S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci.176 (1992) 200-217, Springer, Berlin.
[75]E. Pardoux, F. Pradeilles, Z. Rao, Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Annales de l'Institut Henri Poincare (B) Probability and Statistics 33(4) (1997) 467-490.
[76]E. Pardoux, S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields 114 (1999) 123-150.
[77]E. Pardoux, A. Yu. Veretennikov, Averaging of backward stochastic differential equations, with application to semi-linear PDE's, Stochastics and stochastics re-ports 60 (3-4) (1997) 255-270.
[78]S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports 37 (1991) 61-74.
[79]S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics and Stochastics Reports 38 (1992) 119-134.
[80]S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim.27 (1993) 125-144.
[81]S. Peng, Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control and Optim.37 (1999) 825-843.
[82]S. Peng, M. Xu, The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles, Annale de l'I. H. P., PR 41 (2005) 605-630.
[83]S. Peng, Z. Yang, Anticipated backward stochastic differential equations, The Annals of Probability 37(3) (2009) 877-902.
[84]P. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, second edition,2004.
[85]M. C. Quenez, Methodes de controle stochastique en Finance, These de doctorat de l'Universite Pierre et Marie Curie,1993.
[86]D. Revuz, M. Yor, Continuous Matingales and Brownian Motion,3ed., Springer, New York,1999.
[87]Y. Saisho, Stochastic differential equations for multidimensional domains with reflecting boundary, Prob. Theory and Rel. Fields 74 (1987) 455-477.
[88]Z. Wu, Z. Yu, Fully coupled forward-backward stochastic differential equations and related partial differential equations system, Chinese Journal of Contempo-rary Mathematics 3(2004) 269-282.
[89]Z. Yang, Generalized backward stochastic differential equation. Journal of Shan-dong University (Natural Science) 41(6) (2006) 81-86.
[90]Z. Yang, Anticipated BSDEs and related results in SDEs, Shandong University Doctoral Dissertation, ID:237010200420228,2007, pp.1-72.
[91]J. Yong, Finding adapted solution of forward-backward stochastic differential equations method of continuation, Probab. Theory and Rel. Fields 107 (1997) 537-572.
[92]严加安,彭实戈,方诗赞,吴黎明：随机分析选讲,科学出版社,1997.
[93]严加安：鞅与随机积分引论,上海科学技术出版社,1981.
[94]严加安：测度论讲义,科学出版社,1998.