倒向随机微分方程高精度数值方法
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摘要
本文中我们研究如下形式的倒向随机微分方程(BSDE)：其中T为给定正整数,Wt为定义在概率空间(Ω,F,P，{Ft}0≤t≤T)上的d-维标准布朗运动,f(t,yt,zt)是一个Ft-适应过程(0≤t≤T),ζ是一个{FT}可测的随机变量.12973年,Bismut[36]研究了线性形式的倒向随机微分方程；1990年,Pardoux和Peng[25]给出了BSDE的一般形式,并证明了当生成函数满足Lipschitz条件时倒向随机微分方程(2)解的存在唯一性定理。此后,倒向随机微分方程及其解的形式得到了广泛的研究。在[27]中,Peng证明了正倒向随机微分方程(FBSDE)与偏微分方程之间的直接联系,随后给出了随机最优控制中的一般最大值原理[26]。1997年,N.El Karoui, Peng和Quenez在[15]中通过BSDE获得了推广的Black-Scholes公式,使得BSDE理论逐渐应用于金融理论中,进而使得BSDE理论具有了更大的活力。经过十几年的发展,BSDE理论在随机控制、偏微分方程、金融数学、控制论及经济学等领域得到了广泛的应用。
BSDE在各个领域的应用需要回答的首要问题就是,当给定终端条件和生成函数时如何求解相应的BSDE。但是,我们很难求得一般意义下的BSDE的解析解,目前只有当生成函数为几类特殊的函数时才可以得到其解析解。对于大多数情况,我们只能借助于数值方法来求解BSDE。求解BSDE的数值方法总共可以分成两大类。第一类是基于倒向随机微分方程和偏微分方程之间的关系而提出的数值方法。其中比较有代表性的是Ma, Protter, Yong在[18]中提出的数值求解FBSDE的四步法,Delarue和Menozzi[9]提出的求解全耦合的FBSDE的正倒向随机算法。
第二类方法直接从BSDE本身的特点出发构造数值格式[3；4；5；7；8；9；14；22；29；32；34；35]。当BSDE生成元f不含变量zt时,Chevance[7]提出了基于时空全离散的求解BSDE的数值方法；当BSDE生成元f依赖于变量zt时,Bally[3]提出了基于特殊时间网格的数值方法,避免了对随机积分离散时产生的困难。Bender和Denk提出了求解BSDE的正向格式,Peng在[29]中提出了求解BSDE的线性迭代方法。另外,Memin,Peng和Xu[22]提出了用随机游走模型求解BSDE的数值方法并且给出了收敛性证明。Cvitanic和Zhang[8]给出了应用Monte-Carlo方法求解FBSDE的数值格式,并且在一定的条件下对该格式进行了修正且给出了收敛性分析[32]。在[14]中,Gobet等推广了Zhang的方法,并给出了强Lp(p≥1)意义下的误差估计。Bouchard和Touzi[5]基于Zhang的工作提出了求解BSDE的隐格式。Zhao在[34]中提出了求解BSDE的θ-格式,并且在[35]中给出了误差分析。
我们注意到,现有的绝大部分求解BSDE的数值方法依赖于对标准布朗运动的近似,但是我们无法对标准布朗运动进行高精度的逼近。因此现有的方法都无法实现对BSDE的高精度求解。虽然应用Monte-Carlo方法可以得到较高的精度,但是Monte-Carlo方法的精度1／(?)依赖于随机试验的次数,这使得高精度求解的计算量变得无法接受。目前,θ-格式可以得到较高的精度,但是θ-格式仅仅是一步格式。
本文中我们从以下几个方面研究了求解BSDE的数值方法：
·提出了求解BSDE的多步数值方法。此方法以及全离散的时空网格。在时间坐标轴上,我们用基于多个时间步的Lagrange插值多项式来逼近被积函数,即条件数学期望；在空间坐标轴上,我们用Gauss-Hermite积分公式和多项式插值方法来近似BSDE的解。理论上,只要满足一定的稳定性条件,多步法可以达到任意的精度。
·多步法半离散格式的误差分析。我们证明了当BSDE的生成元,不含变量zt时,多步法的半离散格式收敛性并且证明了半离散格式的收敛阶依赖于求解某一时间层时用到的时间步数。
·多步法的高效率格式。虽然求解BSDE的多步法可以达到很高的精度,但是高精度的要求也使数值求解的计算量变得难以接受。其原因是,当应用Gauss-Hermite积分公式近似条件数学期望时并没有考虑到标准布朗运动的性质。换言之,在构造数值格式的同时,我们应当考虑如何更好的模拟标准布朗运动。因此,我们构造了一类新的离散随机过程,称为Gauss-Hermite过程,并基于此对多步法进行了改进,提出了求解BSDE的高效多步方法。这种方法在使得计算量大大缩减的同时能够与多步法保持完全相同的精度。
·高效多步方法的误差分析。通过倒向追踪高效多步法的误差传播过程,我们得到了如下结论：如果我们依据某些条件对时空网格进行适当比例的缩小,高效多步法与多步法具有完全相同的数值精度,同时计算量得到了大规模的缩小。
·多步法的并行计算。为了进一步提高计算效率,我们研究了多步格式的并行化。在某一时间层求解时,我们将整体的工作量平均分配给若干个进程同时计算,全部完成之后再由控制进程将所有的解收集起来,得到本时间层上的所有数值解。因为多步法和高效多步法均可局部化,所以并行计算可以大大提高求解效率。
论文全文共分五章,具体组织如下：
·第一章：倒向随机微分方程背景知识；
·第二章：BSDE数值方法综述；
·第三章：求解BSDE的多步方法及其误差分析；
·第四章：求解BSDE的高效多步方法、误差分析及并行算法研究；
·第五章：数值试验。
In this thesis, we consider the numerical solution of the following backward stochas-tic differential equation (BSDE): where T is a fixed positive number, Wt is the standard d-dimensional Brownian motion defined on some complete, filtered probability space (Ω,F, P,{Ft}0≤t≤T), f{t,yt,zt) is an adapted stochastic process with respect to {Ft} (0≤t≤T) for each real pair {y,z), andζis an {FT} measurable random variable. The linear form of BSDEs was first introduced in [36]. The existence and uniqueness of the solution of the backward stochastic differential equation (1) are originally proved by Pardoux and Peng in [25] in 1990. Since then BSDEs and their solutions have been extensively studied by many re-searchers. In [27], Peng obtained a direct relation between forward-backward stochastic differential equations(FBSDEs) and partial differential equations, and then in [26], he also found the maximum principle for the stochastic control problems. Many important properties of BSDEs and their applications in finance were studied by Karoui, Peng and Quenez in [15]. An important concept of g-expectation was introduced in [28] for the solutions of BSDEs, and subsequently it was found in [13] that a dynamic coherent risk measure can be valued by a properly defined g-expectation.
While analytic solutions of BSDEs are often very difficult to obtain, approximate numerical solutions of BSDEs are relatively easier to compute and so become highly desired in practical applications. There are mainly two families of numerical methods for solving BSDEs. One is proposed based on the relation between the target BSDE and the corresponding parabolic PDE. A typical four step method for studying FBSDEs was first developed in [18] using this relation. Based on similar ideas, some algorithms were proposed to numerically solve FBSDEs, see [11; 19; 20; 21; 23; 24; 33] for details. When the corresponding parabolic PDE is quasi-linear, Delarue and Menozzi proposed in [9] a special time-space forward-backward discretization scheme based on the theory of fully coupled FBSDEs, and later in [10] they improved the proposed scheme through an interpolation procedure.
The other approach is developed directly based on the target BSDE [3; 4; 5; 7; 8; 9; 14; 22; 29; 32; 34; 35]. In [7], Chevance suggested an effective time-space discretization scheme when the function f does not depend on the variable zt. Bally proposed in [3] a numerical discretization when f depends on the variables yt and zt, where in partic-ular the time discretization is performed on a random net to overcome the difficulties of discretizing the stochastic integral. In [4] Bender and Denk introduced a forward scheme for solving BSDEs. In [29], Peng proposed an iterative linear approximation algorithm which converges under reasonable assumptions. Some random walk methods for solving BSDEs were suggested by Memin, Peng and Xu in [22] and convergences of those methods were also proven there. Cvitanic and Zhang proposed some Monte-Carlo methods for solving FBSDEs in [8], and then under certain weaker regularity assump-tions, Zhang developed a modified numerical scheme and studied its convergence rate in [32]. In [14], Gobet et. al. generalized Zhang's result, and obtained error estimates of the scheme measured in the strong Lp-sense (p≥1). Bouchard and Touzi also pre-sented an implicit scheme in [5] based on Zhang's work. Zhao et. al. proposed an numerical method calledθ-scheme with high accuracy for solving BSDEs in [34] and studied its error estimates in [35].
Note that most of the numerical methods reviewed above for solving BSDEs are not of high accuracy due to the low accuracy of the approximation of the standard Brownian motion. Although the Monte-Carlo method could lead to a good accuracy based on sufficient scale of sampling points, the computational cost is not acceptable because the desired accuracy 1/(?) depends on the number of sampling points N. Theθ-scheme in [34], which is the most accurate scheme so far, is just a one-step method.
In this thesis, we study the numerical methods for solving BSDEs in the following aspects.
●Proposing a stable multi-step method for solving BSDEs. This method is constructed on time-space grids. The integrands, which are conditional mathe-matical expectations derived from the original equation, are approximated by using the Lagrange interpolating polynomials with values of the integrands at multi-time levels. In particular, they are then numerically evaluated using the Gauss-Hermite quadrature rules and polynomial interpolations on the spatial grid. Theoretically, in the sense of multi-step method, a scheme of arbitrary order can be obtained once the conditions are provided.
●Proving the convergence of the semi-discrete scheme. When the generator f is independent of zt, we theoretically prove that the convergence of the semi-discrete scheme can be of high order depending on the number of time levels used.
●Proposing an efficient scheme based on the multi-step scheme. Although the multi-step scheme is of high accuracy, it leads to expensive computation when using large number of time steps. The reason is that we apply Gauss-Hermite quadrature rule to approximate the conditional mathematical expectations with-out considering the properties of the standard Brownian motion. In other words, we don't consider how to construct a discrete stochastic process in our scheme to approximate the behavior of the standard Brownian motion. Therefore, we pro-pose an efficient scheme by constructing a new random walk, called Gauss-Hermite process, based on Gauss-Hermite quadrature rule. By the efficient scheme, the computation expense will be reduced significantly, at the same time, the accuracy of the numerical solution is not affected at all.
●Proving the efficient scheme reaches the same accuracy as the multi-step scheme. By tracking the propagation of the errors in the efficient scheme, we can prove that if the time-space domain is reduced by a suitable proportion, the efficient scheme has the identical accuracy to the multi-step scheme while the computational expense is reduced significantly.
●Parallelization of both schemes. In order to improve the efficiency of the computation, we combine the parallel techniques with our multi-step method in the implementation. Here we divide the whole job on each time level into several sub-jobs according to the number of available processors and then assign each sub-job to a processor and let them do the computations separately. After that, all the results of the sub-jobs are assembled by a control processor. Since our schemes can be easily localized, the computational job for each spatial point at the same time level is independent of other points once the data of the later time levels are ready to use. Thus, our schemes can be parallelized with high efficiency.
The rest of the thesis is organized as follows.
●Chapter 1:Background of backward stochastic differential equations;
●Chapter 2:A brief review of present methods for solving BSDEs:
●Chapter 3:Proposing the multi-step method and obtaining the error estimation of the semi-discrete scheme;
●Chapter 4:Proposing An efficient scheme for solving BSDEs and obtain the error estimation;
●Chapter 5:Numerical Experiments and the investigation into parallelization of both schemes.
BSDE在各个领域的应用需要回答的首要问题就是,当给定终端条件和生成函数时如何求解相应的BSDE。但是,我们很难求得一般意义下的BSDE的解析解,目前只有当生成函数为几类特殊的函数时才可以得到其解析解。对于大多数情况,我们只能借助于数值方法来求解BSDE。求解BSDE的数值方法总共可以分成两大类。第一类是基于倒向随机微分方程和偏微分方程之间的关系而提出的数值方法。其中比较有代表性的是Ma, Protter, Yong在[18]中提出的数值求解FBSDE的四步法,Delarue和Menozzi[9]提出的求解全耦合的FBSDE的正倒向随机算法。
第二类方法直接从BSDE本身的特点出发构造数值格式[3；4；5；7；8；9；14；22；29；32；34；35]。当BSDE生成元f不含变量zt时,Chevance[7]提出了基于时空全离散的求解BSDE的数值方法；当BSDE生成元f依赖于变量zt时,Bally[3]提出了基于特殊时间网格的数值方法,避免了对随机积分离散时产生的困难。Bender和Denk提出了求解BSDE的正向格式,Peng在[29]中提出了求解BSDE的线性迭代方法。另外,Memin,Peng和Xu[22]提出了用随机游走模型求解BSDE的数值方法并且给出了收敛性证明。Cvitanic和Zhang[8]给出了应用Monte-Carlo方法求解FBSDE的数值格式,并且在一定的条件下对该格式进行了修正且给出了收敛性分析[32]。在[14]中,Gobet等推广了Zhang的方法,并给出了强Lp(p≥1)意义下的误差估计。Bouchard和Touzi[5]基于Zhang的工作提出了求解BSDE的隐格式。Zhao在[34]中提出了求解BSDE的θ-格式,并且在[35]中给出了误差分析。
我们注意到,现有的绝大部分求解BSDE的数值方法依赖于对标准布朗运动的近似,但是我们无法对标准布朗运动进行高精度的逼近。因此现有的方法都无法实现对BSDE的高精度求解。虽然应用Monte-Carlo方法可以得到较高的精度,但是Monte-Carlo方法的精度1／(?)依赖于随机试验的次数,这使得高精度求解的计算量变得无法接受。目前,θ-格式可以得到较高的精度,但是θ-格式仅仅是一步格式。
本文中我们从以下几个方面研究了求解BSDE的数值方法：
·提出了求解BSDE的多步数值方法。此方法以及全离散的时空网格。在时间坐标轴上,我们用基于多个时间步的Lagrange插值多项式来逼近被积函数,即条件数学期望；在空间坐标轴上,我们用Gauss-Hermite积分公式和多项式插值方法来近似BSDE的解。理论上,只要满足一定的稳定性条件,多步法可以达到任意的精度。
·多步法半离散格式的误差分析。我们证明了当BSDE的生成元,不含变量zt时,多步法的半离散格式收敛性并且证明了半离散格式的收敛阶依赖于求解某一时间层时用到的时间步数。
·多步法的高效率格式。虽然求解BSDE的多步法可以达到很高的精度,但是高精度的要求也使数值求解的计算量变得难以接受。其原因是,当应用Gauss-Hermite积分公式近似条件数学期望时并没有考虑到标准布朗运动的性质。换言之,在构造数值格式的同时,我们应当考虑如何更好的模拟标准布朗运动。因此,我们构造了一类新的离散随机过程,称为Gauss-Hermite过程,并基于此对多步法进行了改进,提出了求解BSDE的高效多步方法。这种方法在使得计算量大大缩减的同时能够与多步法保持完全相同的精度。
·高效多步方法的误差分析。通过倒向追踪高效多步法的误差传播过程,我们得到了如下结论：如果我们依据某些条件对时空网格进行适当比例的缩小,高效多步法与多步法具有完全相同的数值精度,同时计算量得到了大规模的缩小。
·多步法的并行计算。为了进一步提高计算效率,我们研究了多步格式的并行化。在某一时间层求解时,我们将整体的工作量平均分配给若干个进程同时计算,全部完成之后再由控制进程将所有的解收集起来,得到本时间层上的所有数值解。因为多步法和高效多步法均可局部化,所以并行计算可以大大提高求解效率。
论文全文共分五章,具体组织如下：
·第一章：倒向随机微分方程背景知识；
·第二章：BSDE数值方法综述；
·第三章：求解BSDE的多步方法及其误差分析；
·第四章：求解BSDE的高效多步方法、误差分析及并行算法研究；
·第五章：数值试验。
In this thesis, we consider the numerical solution of the following backward stochas-tic differential equation (BSDE): where T is a fixed positive number, Wt is the standard d-dimensional Brownian motion defined on some complete, filtered probability space (Ω,F, P,{Ft}0≤t≤T), f{t,yt,zt) is an adapted stochastic process with respect to {Ft} (0≤t≤T) for each real pair {y,z), andζis an {FT} measurable random variable. The linear form of BSDEs was first introduced in [36]. The existence and uniqueness of the solution of the backward stochastic differential equation (1) are originally proved by Pardoux and Peng in [25] in 1990. Since then BSDEs and their solutions have been extensively studied by many re-searchers. In [27], Peng obtained a direct relation between forward-backward stochastic differential equations(FBSDEs) and partial differential equations, and then in [26], he also found the maximum principle for the stochastic control problems. Many important properties of BSDEs and their applications in finance were studied by Karoui, Peng and Quenez in [15]. An important concept of g-expectation was introduced in [28] for the solutions of BSDEs, and subsequently it was found in [13] that a dynamic coherent risk measure can be valued by a properly defined g-expectation.
While analytic solutions of BSDEs are often very difficult to obtain, approximate numerical solutions of BSDEs are relatively easier to compute and so become highly desired in practical applications. There are mainly two families of numerical methods for solving BSDEs. One is proposed based on the relation between the target BSDE and the corresponding parabolic PDE. A typical four step method for studying FBSDEs was first developed in [18] using this relation. Based on similar ideas, some algorithms were proposed to numerically solve FBSDEs, see [11; 19; 20; 21; 23; 24; 33] for details. When the corresponding parabolic PDE is quasi-linear, Delarue and Menozzi proposed in [9] a special time-space forward-backward discretization scheme based on the theory of fully coupled FBSDEs, and later in [10] they improved the proposed scheme through an interpolation procedure.
The other approach is developed directly based on the target BSDE [3; 4; 5; 7; 8; 9; 14; 22; 29; 32; 34; 35]. In [7], Chevance suggested an effective time-space discretization scheme when the function f does not depend on the variable zt. Bally proposed in [3] a numerical discretization when f depends on the variables yt and zt, where in partic-ular the time discretization is performed on a random net to overcome the difficulties of discretizing the stochastic integral. In [4] Bender and Denk introduced a forward scheme for solving BSDEs. In [29], Peng proposed an iterative linear approximation algorithm which converges under reasonable assumptions. Some random walk methods for solving BSDEs were suggested by Memin, Peng and Xu in [22] and convergences of those methods were also proven there. Cvitanic and Zhang proposed some Monte-Carlo methods for solving FBSDEs in [8], and then under certain weaker regularity assump-tions, Zhang developed a modified numerical scheme and studied its convergence rate in [32]. In [14], Gobet et. al. generalized Zhang's result, and obtained error estimates of the scheme measured in the strong Lp-sense (p≥1). Bouchard and Touzi also pre-sented an implicit scheme in [5] based on Zhang's work. Zhao et. al. proposed an numerical method calledθ-scheme with high accuracy for solving BSDEs in [34] and studied its error estimates in [35].
Note that most of the numerical methods reviewed above for solving BSDEs are not of high accuracy due to the low accuracy of the approximation of the standard Brownian motion. Although the Monte-Carlo method could lead to a good accuracy based on sufficient scale of sampling points, the computational cost is not acceptable because the desired accuracy 1/(?) depends on the number of sampling points N. Theθ-scheme in [34], which is the most accurate scheme so far, is just a one-step method.
In this thesis, we study the numerical methods for solving BSDEs in the following aspects.
●Proposing a stable multi-step method for solving BSDEs. This method is constructed on time-space grids. The integrands, which are conditional mathe-matical expectations derived from the original equation, are approximated by using the Lagrange interpolating polynomials with values of the integrands at multi-time levels. In particular, they are then numerically evaluated using the Gauss-Hermite quadrature rules and polynomial interpolations on the spatial grid. Theoretically, in the sense of multi-step method, a scheme of arbitrary order can be obtained once the conditions are provided.
●Proving the convergence of the semi-discrete scheme. When the generator f is independent of zt, we theoretically prove that the convergence of the semi-discrete scheme can be of high order depending on the number of time levels used.
●Proposing an efficient scheme based on the multi-step scheme. Although the multi-step scheme is of high accuracy, it leads to expensive computation when using large number of time steps. The reason is that we apply Gauss-Hermite quadrature rule to approximate the conditional mathematical expectations with-out considering the properties of the standard Brownian motion. In other words, we don't consider how to construct a discrete stochastic process in our scheme to approximate the behavior of the standard Brownian motion. Therefore, we pro-pose an efficient scheme by constructing a new random walk, called Gauss-Hermite process, based on Gauss-Hermite quadrature rule. By the efficient scheme, the computation expense will be reduced significantly, at the same time, the accuracy of the numerical solution is not affected at all.
●Proving the efficient scheme reaches the same accuracy as the multi-step scheme. By tracking the propagation of the errors in the efficient scheme, we can prove that if the time-space domain is reduced by a suitable proportion, the efficient scheme has the identical accuracy to the multi-step scheme while the computational expense is reduced significantly.
●Parallelization of both schemes. In order to improve the efficiency of the computation, we combine the parallel techniques with our multi-step method in the implementation. Here we divide the whole job on each time level into several sub-jobs according to the number of available processors and then assign each sub-job to a processor and let them do the computations separately. After that, all the results of the sub-jobs are assembled by a control processor. Since our schemes can be easily localized, the computational job for each spatial point at the same time level is independent of other points once the data of the later time levels are ready to use. Thus, our schemes can be parallelized with high efficiency.
The rest of the thesis is organized as follows.
●Chapter 1:Background of backward stochastic differential equations;
●Chapter 2:A brief review of present methods for solving BSDEs:
●Chapter 3:Proposing the multi-step method and obtaining the error estimation of the semi-discrete scheme;
●Chapter 4:Proposing An efficient scheme for solving BSDEs and obtain the error estimation;
●Chapter 5:Numerical Experiments and the investigation into parallelization of both schemes.
引文
[1]EDWARD OMBERG, Efficient Discrete Time Jump Process Models in Option Pric-ing, J. Financial and Quantitative Analysis,23(1988), pp.161-174
[2]M. ABRAMOWITZ AND I. STEGUN, Handbook of Mathematical Functions, Dover Publications, New York,1972.
[3]V. BALLY, An approximation scheme for BSDEs and applications to control and nonlinear PDEs, Pitman Research Notes in Mathematics Series, Longman, 364(1997).
[4]C. BENDER AND R. DENK, A forward scheme for backward SDEs, Stochastic Process. Appl.,117(2007), pp.1793-1812.
[5]B. BOUCHARD AND N. TOUZI, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications,111(2004), pp.175-206.
[6]J. BUTCHER, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd,2003.
[7]D. CHEVANCE, Numerical Methods for Backward Stochastic Differential Equa-tions, in Numerical Methods in Finance, Edited by L. C. G. Rogers and D. Talay, Cambridge University Press,1997, pp.232(?)244.
[8]J. CVITANIC AND J. ZHANG, The Steepest Descent Method for FBSDEs, Elec-tronic Journal of Probability,10(2005), pp, 1468-1495.
[9]F. DELARUE AND S. MENOZZI, A forward-backward stochastic algorithm for quasi-linear PDEs, The Annals of Applied Probability,16(2006), pp.140-184.
[10]F. DELARUE AND S. MENOZZI, An interpolated stochastic algorithm for quasi-linear PDEs, Mathematics of Computation,77(2008), pp.125-158.
[11]J. DOUGLAS JR., J. MA, AND P. PROTTER, Numerical methods for forward-backward stochastic differential equations, The Annals of Applied Probability, 6(1996), pp.940-968.
[12]E. HAIRER AND G. WANNER, Solving Ordinary Dfferential Equation Ⅰ, Ⅱ, Springer-verlag Berlin Heidelberg,1996.
[13]E. R. GIANIN, Risk measures via g-expectations, Insurance:Mathematics and Economics,39(2006), pp.19-34.
[14]E. GOBET AND C. LABART, Error expansion for the discretization of Backward Stochastic Differential Equations, Preprint submitted to Stochastic Processes and their Applications,17th February,2006.
[15]N. EL. KAROUI, S. PENG, AND M. C. QUENEZ, Backward stochastic differential equations in finance, Mathematical Finance,7(1997), pp.1-71.
[16]O. LADYZENSKAJA, O. SOLONNIKOV, N. URALCEVA, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Math. Monographs,23, AMS, Prov-idence Rhode Island,1968.
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[18]J. MA, P. PROTTER, AND J. YONG, Solving forward-backward stochastic dif-ferential equations explicitly-a four step scheme, Probability Theory and Related Fields,98(1994), pp.339-359.
[19]J. MA AND J. YONG, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics,1702, Springer,1999.
[20]J. MA AND J. ZHANG, Representation theorems for backward stochastic differ-ential equations, The Annals of Applied Probability,12(4)(2002), pp.1390-1418.
[21]J. MA, J. SHEN AND Y. ZHAO, On Numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal.,46(5) (2008), pp.2636-2661.
[22]J. MEMIN, S. PENG, AND M. Xu, Convergence of solutions of discrete reflected backward SDE's and simulations, preprint of Institut de Recherche Mathematique de Rennes, Rennes,2003.
[23]G. MILSTEIN AND M. TRETYAKOV, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput.,28(2006),561-582.
[24]G. MILSTEIN AND M. TRETYAKOV, Discretization of forward-backward stochas-tic differential equations and related quasi-linear parabolic equations, IMA J. Nu-mer. Anal.,27(2007),24-44.
[25]E. PARDOUX AND S. PENG. Adapted solution of a backward stochastic differntial equation, Systems and Control Letters,14(1990), pp.55-61.
[26]S. PENG, A general stochastic maximum principle for optimal control problems, SIAM J. Control and Optimization,28(1990), pp.966-979.
[27]S. PENG, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports,37(1991), pp.61-74.
[28]S. PENG, Backward SDE and Related g-Expectation, Backward Stochastic Differ-ential Equations, Pitman Research Notes in Math. Series,1997, No.364,141-159
[29]S. PENG, A linear approximation algorithm using BSDE, Pacific Economic Re-view,4(1999), pp.285-291.
[30]G. DA PRATO AND J. ZABCZYK, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press,2002.
[31]E. SULI AND D. MAYERS, An Introduction to Numerical Analysis, Cambridge University Press,2003.
[32]J. ZHANG, A numerical scheme for BSDEs, Ann. Appl. Probab.,14(2004), pp. 459-488.
[33]Y. ZHANG AND W. ZHENG, Discretizing a backward stochastic differential equa-tion, Inter. J. Mathematics and Mathematical Sciences,32(2002), pp.103-116.
[34]W. ZHAO, L. CHEN, AND S. PENG, A new kind of accurate numerical method for backward stochastic differential equations, Siam J. Sci. Comput.,28(2006), pp. 1563-1581.
[35]W. ZHAO, J. WANG, AND S. PENG, Error Estimates of the θ-Scheme for Back-ward Stochastic Differential Equations, Dis. Cont. Dyn. Sys. B,12(2009), pp. 905-924.
[36]Bismut, J. (1973). Conjugate convex functions in optimal stochastic control. Jour-nal of Mathematical Analysis and Applications,44,384-404.
[37]Black, F.& Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy,81,637-654.
[38]Briand, P., Coquet, F., Hu, Y., Memin, J.& Peng, S. (2000). A converse compar-ison theorem for BSDEs and related properties of g-expectation. Elect. Comm. in Probab.,5,101-117.
[39]Briand, P., Delyon, B. & Memin, J. (2001). Donsker-type theorem for BSDEs. Elect. Comm. in Probab.,6,1-14.
[40]F. Coquet, V. Mackevicius,and J.Memin, Stability in D of martingales and back-ward equations under perturbation of filtrations, Stochastic Process. Appl.,75(2), p.235-248,1998.
[41]Crandall, M.G., Ishii, H.& Lions, P.L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society,27,1-67.
[42]Duffie, D.& Epstein, L.G. (1992). Stochastic differential utility. Econometrica, 60,353-394, appendix C with Costis Skiadas.
[43]El Karoui, N., Jeanblanc-Picque, M.& Shreve, S.E. (1998). Robustness of the Black and Scholes formula. Mathematical Finance,8,93-126.
[44]Emmanuel Gobet, Jean-Philippe Lemor and Xavier Warin, A REGRESSION-BASED MONTE CARLO METHOD TO SOLVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS The Annals of Applied Probability 2005, Vol.15, No.3,2172-2202.
[45]Jean-Philippe Lemor, Emmanuel Gobet and Xavier Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli Volume 12, Number 5 (2006),889-916.
[46]F. Longstaff, E. Schwartz,Valuing American options by simulation:a least squares. approach. Review of Financial Studies(2001),14:113-147,2001.
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[10]F. DELARUE AND S. MENOZZI, An interpolated stochastic algorithm for quasi-linear PDEs, Mathematics of Computation,77(2008), pp.125-158.
[11]J. DOUGLAS JR., J. MA, AND P. PROTTER, Numerical methods for forward-backward stochastic differential equations, The Annals of Applied Probability, 6(1996), pp.940-968.
[12]E. HAIRER AND G. WANNER, Solving Ordinary Dfferential Equation Ⅰ, Ⅱ, Springer-verlag Berlin Heidelberg,1996.
[13]E. R. GIANIN, Risk measures via g-expectations, Insurance:Mathematics and Economics,39(2006), pp.19-34.
[14]E. GOBET AND C. LABART, Error expansion for the discretization of Backward Stochastic Differential Equations, Preprint submitted to Stochastic Processes and their Applications,17th February,2006.
[15]N. EL. KAROUI, S. PENG, AND M. C. QUENEZ, Backward stochastic differential equations in finance, Mathematical Finance,7(1997), pp.1-71.
[16]O. LADYZENSKAJA, O. SOLONNIKOV, N. URALCEVA, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Math. Monographs,23, AMS, Prov-idence Rhode Island,1968.
[17]J. MA, P.PROTTER, J. SAN MARTIN, AND S. TORRES, Numerical meth-ods for backward stochastic differential equations, Annals of Applied Probability, 12(2002), pp.302-316.
[18]J. MA, P. PROTTER, AND J. YONG, Solving forward-backward stochastic dif-ferential equations explicitly-a four step scheme, Probability Theory and Related Fields,98(1994), pp.339-359.
[19]J. MA AND J. YONG, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics,1702, Springer,1999.
[20]J. MA AND J. ZHANG, Representation theorems for backward stochastic differ-ential equations, The Annals of Applied Probability,12(4)(2002), pp.1390-1418.
[21]J. MA, J. SHEN AND Y. ZHAO, On Numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal.,46(5) (2008), pp.2636-2661.
[22]J. MEMIN, S. PENG, AND M. Xu, Convergence of solutions of discrete reflected backward SDE's and simulations, preprint of Institut de Recherche Mathematique de Rennes, Rennes,2003.
[23]G. MILSTEIN AND M. TRETYAKOV, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput.,28(2006),561-582.
[24]G. MILSTEIN AND M. TRETYAKOV, Discretization of forward-backward stochas-tic differential equations and related quasi-linear parabolic equations, IMA J. Nu-mer. Anal.,27(2007),24-44.
[25]E. PARDOUX AND S. PENG. Adapted solution of a backward stochastic differntial equation, Systems and Control Letters,14(1990), pp.55-61.
[26]S. PENG, A general stochastic maximum principle for optimal control problems, SIAM J. Control and Optimization,28(1990), pp.966-979.
[27]S. PENG, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports,37(1991), pp.61-74.
[28]S. PENG, Backward SDE and Related g-Expectation, Backward Stochastic Differ-ential Equations, Pitman Research Notes in Math. Series,1997, No.364,141-159
[29]S. PENG, A linear approximation algorithm using BSDE, Pacific Economic Re-view,4(1999), pp.285-291.
[30]G. DA PRATO AND J. ZABCZYK, Second Order Partial Differential Equations in Hilbert Spaces, Cambridge University Press,2002.
[31]E. SULI AND D. MAYERS, An Introduction to Numerical Analysis, Cambridge University Press,2003.
[32]J. ZHANG, A numerical scheme for BSDEs, Ann. Appl. Probab.,14(2004), pp. 459-488.
[33]Y. ZHANG AND W. ZHENG, Discretizing a backward stochastic differential equa-tion, Inter. J. Mathematics and Mathematical Sciences,32(2002), pp.103-116.
[34]W. ZHAO, L. CHEN, AND S. PENG, A new kind of accurate numerical method for backward stochastic differential equations, Siam J. Sci. Comput.,28(2006), pp. 1563-1581.
[35]W. ZHAO, J. WANG, AND S. PENG, Error Estimates of the θ-Scheme for Back-ward Stochastic Differential Equations, Dis. Cont. Dyn. Sys. B,12(2009), pp. 905-924.
[36]Bismut, J. (1973). Conjugate convex functions in optimal stochastic control. Jour-nal of Mathematical Analysis and Applications,44,384-404.
[37]Black, F.& Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy,81,637-654.
[38]Briand, P., Coquet, F., Hu, Y., Memin, J.& Peng, S. (2000). A converse compar-ison theorem for BSDEs and related properties of g-expectation. Elect. Comm. in Probab.,5,101-117.
[39]Briand, P., Delyon, B. & Memin, J. (2001). Donsker-type theorem for BSDEs. Elect. Comm. in Probab.,6,1-14.
[40]F. Coquet, V. Mackevicius,and J.Memin, Stability in D of martingales and back-ward equations under perturbation of filtrations, Stochastic Process. Appl.,75(2), p.235-248,1998.
[41]Crandall, M.G., Ishii, H.& Lions, P.L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society,27,1-67.
[42]Duffie, D.& Epstein, L.G. (1992). Stochastic differential utility. Econometrica, 60,353-394, appendix C with Costis Skiadas.
[43]El Karoui, N., Jeanblanc-Picque, M.& Shreve, S.E. (1998). Robustness of the Black and Scholes formula. Mathematical Finance,8,93-126.
[44]Emmanuel Gobet, Jean-Philippe Lemor and Xavier Warin, A REGRESSION-BASED MONTE CARLO METHOD TO SOLVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS The Annals of Applied Probability 2005, Vol.15, No.3,2172-2202.
[45]Jean-Philippe Lemor, Emmanuel Gobet and Xavier Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli Volume 12, Number 5 (2006),889-916.
[46]F. Longstaff, E. Schwartz,Valuing American options by simulation:a least squares. approach. Review of Financial Studies(2001),14:113-147,2001.