有连续下界的反射重随机微分方程的数值计算
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摘要
Feynman-Kac公式解决了二阶线抛物型和椭圆型偏微分方程的显式解问题。为了推广Feynman-Kac公式,得到次线性偏微分方程的解,Pardoux和彭在[1]中建立并研究了一类新的随机方程：倒向随机微分方程(BDSDEs)。伴随着Feynman-Kac公式的扩展形式(给出了二阶线抛物型和椭圆型随机偏微分方程的显示解),Pardoux和彭继而在[4]中引入了一类新的方程：倒向重随机微分方程(BDSDEs),给出了次线性随机偏微分方程的显示解。
相应的,石玉峰和谷燕玲在研究带边界的随机偏微分方程过程中引入了带反射的倒向重随机微分方程(RBDSDEs)[29],并给出了其解的存在唯一性条件。Auguste Aman及Modeste N'Zi在[5]中也完成了类似工作。
带反射边界的倒向重随机微分方程在存在各种限制条件的经济模型中有很好的应用,但是绝大部分的带反射倒向重随机微分方程是没有显示解的。所以给出其数值解是十分必要和有用的。本文用惩罚的办法,通过将布朗运动近似为伯努利序列逼近来求得带连续下边界的倒向重随机微分方程的数值解,并证明解的收敛性。
考虑下述下边界的重随机反射微分方程
其中,Bt和Wt为相互独立的标准布朗运动。Kt是不降过程且K0=O。将[0,T]n等分,记这里1≤J≤n。我们定义如下的过程来近似标准布朗运动Bt,为两列独立的伯努利序列。且
考虑区间[tj,tj+1],通过上述替换我们得到RBDSDEs的离散近似
通过惩罚方法,用来逼近我们得到
通过取条件期望的办法,我们可以得到上述方程的隐格式解。很多情况下,为了取得显式解,我们将方程中f(·,·,·)中的yjp,n替换为用同样的方法可以得到方程的显格式解。
基于信息族的弱收敛以及重随机微分方程中数值算法收敛性的证明,我们可以类似的得到反射方程数值解的收敛证明。
本文结构如下：
第’章介绍了倒向重随机方程及其数值算法的研究现状。
第二章介绍了带下边界的反射倒向重随机微分方程的基本理论。
第三章利用伯努利序列来近似布朗运动,在惩罚方法的基础上给出了下边界的倒向重随机微分方程的显格式与隐格式的数值算法。
第四章在引用倒向重随机方程的信息族收敛性结果以及扩展的Donskcr-Type定理的基础上,证明了显格式与隐格式算法的收敛性。
Since Non-linear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], the theory designed by this paper has been found to be powerful tool for obtaining the probabilistic in-terpretation of partial differential equation (PDE for short) while backward doubly stochastic differential equations (BDSDEs for short) can obtain the probabilistic interpretation of stochastic PDEs.
On the other hand in order to give a probabilistic interpretation to the particularly class of stochastic PDEs, the solution of which has some limits, Yufeng Shi and Yanling Gu [29] introduced a new kind of equation:reflected backward doubly stochastic differential equations (RBDSDEs for short). Au-guste Aman and Modeste N'Zi [5] have done the similar research.
However, RBDSDEs can not be solved explicitly. To develop numerical method and numerical algorithm is very helpful,theoretically and practically.
In this paper we study two numerical forms (implicit explicit) of approx-imating solutions of reflected backward doubly stochastic differential equa-tions (RBDSDEs for short) with one lower continuous barrier and prove its convergence.
Consider the following RBDSDEs with one lower barrier
where Bt and Wt are two mutually independent standard Brown motion processes. Kt is an increasing process and K0 = 0. We divide the time in-terval [O,T] into n parts:O= t0 On the small interval [tj,tj+1], the equation above can be writen as
By penalization method, replace we have
Taking conditional expectation, We solve the equation above and obtain the implicit solution. In fact, ytp,n can not be solved explicitly in many cases which f nonlinearly depends on y. In this case, we apply the explicit penaliza-tion scheme replacing yJp,n in f by E[(?)].
Based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme.
The paper is organized as follows:
The first chapter introduces the current research of RBDSDEs and its nu-merical computation.
In chapter two, we offer some preliminaries of RBDSDEs.
In chapter three, by replacing Bernoulli sequences and penalization method, we consturct the implicit and explicit scheme of RBDSDEs.
In chapter fore, based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme.
相应的,石玉峰和谷燕玲在研究带边界的随机偏微分方程过程中引入了带反射的倒向重随机微分方程(RBDSDEs)[29],并给出了其解的存在唯一性条件。Auguste Aman及Modeste N'Zi在[5]中也完成了类似工作。
带反射边界的倒向重随机微分方程在存在各种限制条件的经济模型中有很好的应用,但是绝大部分的带反射倒向重随机微分方程是没有显示解的。所以给出其数值解是十分必要和有用的。本文用惩罚的办法,通过将布朗运动近似为伯努利序列逼近来求得带连续下边界的倒向重随机微分方程的数值解,并证明解的收敛性。
考虑下述下边界的重随机反射微分方程
其中,Bt和Wt为相互独立的标准布朗运动。Kt是不降过程且K0=O。将[0,T]n等分,记这里1≤J≤n。我们定义如下的过程来近似标准布朗运动Bt,为两列独立的伯努利序列。且
考虑区间[tj,tj+1],通过上述替换我们得到RBDSDEs的离散近似
通过惩罚方法,用来逼近我们得到
通过取条件期望的办法,我们可以得到上述方程的隐格式解。很多情况下,为了取得显式解,我们将方程中f(·,·,·)中的yjp,n替换为用同样的方法可以得到方程的显格式解。
基于信息族的弱收敛以及重随机微分方程中数值算法收敛性的证明,我们可以类似的得到反射方程数值解的收敛证明。
本文结构如下：
第’章介绍了倒向重随机方程及其数值算法的研究现状。
第二章介绍了带下边界的反射倒向重随机微分方程的基本理论。
第三章利用伯努利序列来近似布朗运动,在惩罚方法的基础上给出了下边界的倒向重随机微分方程的显格式与隐格式的数值算法。
第四章在引用倒向重随机方程的信息族收敛性结果以及扩展的Donskcr-Type定理的基础上,证明了显格式与隐格式算法的收敛性。
Since Non-linear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], the theory designed by this paper has been found to be powerful tool for obtaining the probabilistic in-terpretation of partial differential equation (PDE for short) while backward doubly stochastic differential equations (BDSDEs for short) can obtain the probabilistic interpretation of stochastic PDEs.
On the other hand in order to give a probabilistic interpretation to the particularly class of stochastic PDEs, the solution of which has some limits, Yufeng Shi and Yanling Gu [29] introduced a new kind of equation:reflected backward doubly stochastic differential equations (RBDSDEs for short). Au-guste Aman and Modeste N'Zi [5] have done the similar research.
However, RBDSDEs can not be solved explicitly. To develop numerical method and numerical algorithm is very helpful,theoretically and practically.
In this paper we study two numerical forms (implicit explicit) of approx-imating solutions of reflected backward doubly stochastic differential equa-tions (RBDSDEs for short) with one lower continuous barrier and prove its convergence.
Consider the following RBDSDEs with one lower barrier
where Bt and Wt are two mutually independent standard Brown motion processes. Kt is an increasing process and K0 = 0. We divide the time in-terval [O,T] into n parts:O= t0
By penalization method, replace we have
Taking conditional expectation, We solve the equation above and obtain the implicit solution. In fact, ytp,n can not be solved explicitly in many cases which f nonlinearly depends on y. In this case, we apply the explicit penaliza-tion scheme replacing yJp,n in f by E[(?)].
Based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme.
The paper is organized as follows:
The first chapter introduces the current research of RBDSDEs and its nu-merical computation.
In chapter two, we offer some preliminaries of RBDSDEs.
In chapter three, by replacing Bernoulli sequences and penalization method, we consturct the implicit and explicit scheme of RBDSDEs.
In chapter fore, based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme.
引文
[1]Pardoux E. and Peng S. Adapted solutions of a backward stochastics dif-ferential equation. Systems Control Letters,1990,14:55-61.
[2]Pardoux E. and Peng S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In:B.L.Rozovski.R.B. Sowers(eds). Stochastic partial equations and their applications. Lect. Notes control Inf. Sci.176,200-217, Springer, Berlin.1992.
[3]Peng S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,Stochastics 37:61-74,1991.
[4]Pardoux E. and Peng S. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat Fields,98,209-227,1994.
[5]Auguste Aman and Modeste N'Zi, Reflected backward doubly stochastic differential equations and application,UFR de Mathematiques et Informa-tique,22 BP 582 Abidjan 22, Cote d'Ivoirem2010.
[6]Shi Y. Yang W. and Yuan J. Numerical Computation for Backward Doubly SDEd and SPDEs,2008.
[7]Lepeltier. J. P. and Xu. M. Penalization method for Reflected Backward Stochastic Differential Equations with one r.c.1.1. barrier. Statistics and Probability Letters,75,58-66,2005.
[8]Ma J. Protter P. San Martin J. and Torres S. Numerical method for back-ward stochastic differential equations, Ann. Appl. Probab.12, no.1,302-316,2002
[9]Memin J. Peng S. and Xu M. Convergence of solutions of discrete Reflect-ed backward SDEs and simulations, Acta Mathematicae Sinica (English Series), Vol.24, No.1,1-18,2008
[10]Frangois Coquet, Jean Memin and Leszek Slominski, On weak conver-gence of filtrations, Campus de Beaulieu.35042 Rennes Cedex, Prance, X. Copernicus University, u1. Chopina,87-100 Torun, Poland.2001.
[11]Philippe Briand, Bernard Delyon, Jean Memin, Donsker-Type theorem for BSDEs, Elect. Comm. in Probab.6.1-14,2001.
[12]Memin J. Peng S. and Xu M. Numerical Algorithms for BSDEs:Conver-gence and simulation, arXiv:math.PR/0611864v2.
[13]Zhang Y. and Zheng W. Discretizing a backward stochastic differential equation, Int J. Math. Sci.32, no.2,103-116,2002.
[14]Philippe Briand, Bernard Delyon, Jean Memin, On the robustness of back-ward stochastic differential equations, Stochastic Processes and their Ap-plications 97,229-253,2002.
[15]Xu M. Numerical Algorithms and Simulations for Reflected Backward Stochastic Differential Equations with two Continuous Barriers, Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,2008
[16]Bellalah M. and Wu Z. A simple model of corporate international in-vestment under incomplete information and taxes. Springer Science and Bussiness Media, LLC 2008.
[17]El Karoui N., Peng S. and Quenez M.C. Backward Stochastic Differential Equations In Finance. Mathematical Finance,1997, Vol.7, No.1:1-17.
[18]彭实戈.倒向随机微分方程及其应用.数学进展,1997,26(2)：97-111.
[19]周少甫,黄志远,张子刚.倒向随机微分方程的理论、发展及其应用.应用数学,2002,15：9-13.
[20]Peng S. Backward stochastic diferential equation and application to opti-mal control, Appl. Math. Optim,1993,127:125-144.
[21]D.Aldous,Weak convergence for stochastic processes for processes viewed in the Strasbourg manner, Preprint, Statist. Laboratory Univ. Cambridge, 1978.
[22]D.Revuz and M.Yor, Continuous Martingales and Brownian Motion, Springer Verlag, Berlin Heidelberg Now-York,1991.
[23]Barles G. Buckdahn R. Pardoux E. Backward stochastic differential equa-tions and integral-partial differential equations. Stochastics Rep,60.57-83. 1997.
[24]D.Chevance, Numerical methods for backward stochastic differential e-quations, Numerical methods in finance, Cambridge Univ. Predd, Cam-bridge, pp.232-244,1997.
[25]N. El Karoui, S.peng, and M.-C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7, no.1,1-71.1997
[26]Kloeden P.E. and Platen E. Numerical solution of stochastic differential equations. Springer. Berlin.1992.
[27]Zhang J.2004. A Numerical scheme for backward stochastic equations. The Annals of Applied Probability.14(1),459-48.2004.
[28]Yang, W.Q. Numerical Computation Examples for Backward Doubly S-DEs, http://finance.math.sdu.edu.cn/faculty/yang/index.htm.2008.
[29]Yufeng Shi, Yanling Gu, Reflected Solutions of Backward Doubly Stochas-tic Differential Equations. arXiv:0806.0917v2 [math. PR],2009.
[2]Pardoux E. and Peng S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In:B.L.Rozovski.R.B. Sowers(eds). Stochastic partial equations and their applications. Lect. Notes control Inf. Sci.176,200-217, Springer, Berlin.1992.
[3]Peng S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,Stochastics 37:61-74,1991.
[4]Pardoux E. and Peng S. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat Fields,98,209-227,1994.
[5]Auguste Aman and Modeste N'Zi, Reflected backward doubly stochastic differential equations and application,UFR de Mathematiques et Informa-tique,22 BP 582 Abidjan 22, Cote d'Ivoirem2010.
[6]Shi Y. Yang W. and Yuan J. Numerical Computation for Backward Doubly SDEd and SPDEs,2008.
[7]Lepeltier. J. P. and Xu. M. Penalization method for Reflected Backward Stochastic Differential Equations with one r.c.1.1. barrier. Statistics and Probability Letters,75,58-66,2005.
[8]Ma J. Protter P. San Martin J. and Torres S. Numerical method for back-ward stochastic differential equations, Ann. Appl. Probab.12, no.1,302-316,2002
[9]Memin J. Peng S. and Xu M. Convergence of solutions of discrete Reflect-ed backward SDEs and simulations, Acta Mathematicae Sinica (English Series), Vol.24, No.1,1-18,2008
[10]Frangois Coquet, Jean Memin and Leszek Slominski, On weak conver-gence of filtrations, Campus de Beaulieu.35042 Rennes Cedex, Prance, X. Copernicus University, u1. Chopina,87-100 Torun, Poland.2001.
[11]Philippe Briand, Bernard Delyon, Jean Memin, Donsker-Type theorem for BSDEs, Elect. Comm. in Probab.6.1-14,2001.
[12]Memin J. Peng S. and Xu M. Numerical Algorithms for BSDEs:Conver-gence and simulation, arXiv:math.PR/0611864v2.
[13]Zhang Y. and Zheng W. Discretizing a backward stochastic differential equation, Int J. Math. Sci.32, no.2,103-116,2002.
[14]Philippe Briand, Bernard Delyon, Jean Memin, On the robustness of back-ward stochastic differential equations, Stochastic Processes and their Ap-plications 97,229-253,2002.
[15]Xu M. Numerical Algorithms and Simulations for Reflected Backward Stochastic Differential Equations with two Continuous Barriers, Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,2008
[16]Bellalah M. and Wu Z. A simple model of corporate international in-vestment under incomplete information and taxes. Springer Science and Bussiness Media, LLC 2008.
[17]El Karoui N., Peng S. and Quenez M.C. Backward Stochastic Differential Equations In Finance. Mathematical Finance,1997, Vol.7, No.1:1-17.
[18]彭实戈.倒向随机微分方程及其应用.数学进展,1997,26(2)：97-111.
[19]周少甫,黄志远,张子刚.倒向随机微分方程的理论、发展及其应用.应用数学,2002,15：9-13.
[20]Peng S. Backward stochastic diferential equation and application to opti-mal control, Appl. Math. Optim,1993,127:125-144.
[21]D.Aldous,Weak convergence for stochastic processes for processes viewed in the Strasbourg manner, Preprint, Statist. Laboratory Univ. Cambridge, 1978.
[22]D.Revuz and M.Yor, Continuous Martingales and Brownian Motion, Springer Verlag, Berlin Heidelberg Now-York,1991.
[23]Barles G. Buckdahn R. Pardoux E. Backward stochastic differential equa-tions and integral-partial differential equations. Stochastics Rep,60.57-83. 1997.
[24]D.Chevance, Numerical methods for backward stochastic differential e-quations, Numerical methods in finance, Cambridge Univ. Predd, Cam-bridge, pp.232-244,1997.
[25]N. El Karoui, S.peng, and M.-C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7, no.1,1-71.1997
[26]Kloeden P.E. and Platen E. Numerical solution of stochastic differential equations. Springer. Berlin.1992.
[27]Zhang J.2004. A Numerical scheme for backward stochastic equations. The Annals of Applied Probability.14(1),459-48.2004.
[28]Yang, W.Q. Numerical Computation Examples for Backward Doubly S-DEs, http://finance.math.sdu.edu.cn/faculty/yang/index.htm.2008.
[29]Yufeng Shi, Yanling Gu, Reflected Solutions of Backward Doubly Stochas-tic Differential Equations. arXiv:0806.0917v2 [math. PR],2009.