带停时的倒向重随机微分方程
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摘要
在本文中，我们给出了一类带停时的倒向重随机微分方程(BDSDEs)的参数的一个充分条件，在这个条件下，对于任意平方可积的随机变量，带停时的倒向重随机微分方程存在唯一的解。同时，我们讨论了这类方程的连续依赖性和连续收敛性定理。
自从Pardoux和Peng在1990年引入了倒向随机微分方程(简称BSDEs)以来，倒向随机微分方程理论得到了迅速的发展和广泛的应用。主要结论是，对于任意给定的ξ∈L~2(Ω，F_T，P；R~k)，如下的倒向随机微分方程在一定条件下在有限时间区间[0，T]上存在唯一的解(y_t，z_t)。这类方程为准确描述许多金融数学中的问题提供了一个有用的框架，同时对于解决随机控制，随机微分，以及拟线性偏微分方程解的概率表示中的问题有很大的帮助。
在1990年引入了倒向随机微分方程理论以后，Pardoux和Peng在1994年引入了一类新的倒向随机微分方程-倒向重随机微分方程，并证明了有限时间区间上的倒向重随机微分方程的解的存在性和唯一性定理。这些研究成果使我们能够给出一类拟线性随机偏微分方程解的概率表示，从而扩展了线性随机偏微分方程的Feynman-Kac公式。在[3]中，Chen和Wang把倒向随机微分方程从有限时间区间推广到了无限时间区间。在[4]中，Chen给出并证明了带停时的倒向随机微分方程解的存在唯一性定理。我们认为带停时的倒向重随机微分方程也能给出一类拟线性随机偏微分方程的解的概率表示。仿照Chen和Wang的方法，我们给出了带停时的倒向重随机微分方程的解的存在唯一性定理，并证明了连续依赖性和连续收敛性定理。
本文的结构如下：在第二章中，我们列出了在这篇文章中用到的几个重要的定理。在第三章中，我们给出了所要研究问题的基本设定和主要假设。第四章，我们首先在第一节中提出并证明了文章的主要结果-存在唯一性定理，然后在第二节中讨论了解的连续依赖性和收敛性定理。
In this paper, we give a sufficient condition on the coefficients of a class of backward doubly stochastic differential equations with stopping time (BDSDEs) under which the BDSDEs with stopping time have a unique solution for any given square integrable terminal values. We also discuss the continuous dependence theorem and convergence theorem for this class of equations.
Since the backward stochastic differential equations (BSDEs in short) was introduced by Pardoux and Peng in 1990, the theory of BSDEs has been developed and widely used by many researchers. The main result is, for any givenξ∈L~2(Ω, F_T, P; R~k) , the following BSDEhas a unique solution pair (y_t,z_t) in the interval [0, T] under some condition. This class of equations provide a useful framework for formulating many problems in mathematical finance, and they are also useful for problems in stochastic control, stochastic differential game, and probabilistic formula for the solutions of quasi-linear partial differential equations.
After Pardoux and Peng (1990) introduced the theory of BSDEs, Pardoux and Peng introduced a new class of backward stochastic differential equations-backward "doubly" stochastic differential equations and also showed the existence and uniqueness theorem of the solution of BDSDEs in a finite time interval. These research allow us to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDEs. In [3], Chen and Wang extended BSDEs from finite time interval [0,T] to infinite time interval [0,∞]. Chen also proved the existence and uniqueness theorem for BSDEs with stopping time in [4]. BDSDEs with stopping time are also very interesting to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations. Here following the approach of Chen and Wang, we present the existence and uniqueness theorem for BDSDEs with stopping time and proved the continuous dependence theorem and convergence theorem.
This paper is organized as follows. In chapter 2, we show some important theorems we used in our paper. In chapter 3, we present the setting of the problem and the main assumptions. In chapter 4, first we present and prove the main result-the existence and uniqueness theorem. Then we discuss the continuous dependence theorem and convergence theorem.
自从Pardoux和Peng在1990年引入了倒向随机微分方程(简称BSDEs)以来，倒向随机微分方程理论得到了迅速的发展和广泛的应用。主要结论是，对于任意给定的ξ∈L~2(Ω，F_T，P；R~k)，如下的倒向随机微分方程在一定条件下在有限时间区间[0，T]上存在唯一的解(y_t，z_t)。这类方程为准确描述许多金融数学中的问题提供了一个有用的框架，同时对于解决随机控制，随机微分，以及拟线性偏微分方程解的概率表示中的问题有很大的帮助。
在1990年引入了倒向随机微分方程理论以后，Pardoux和Peng在1994年引入了一类新的倒向随机微分方程-倒向重随机微分方程，并证明了有限时间区间上的倒向重随机微分方程的解的存在性和唯一性定理。这些研究成果使我们能够给出一类拟线性随机偏微分方程解的概率表示，从而扩展了线性随机偏微分方程的Feynman-Kac公式。在[3]中，Chen和Wang把倒向随机微分方程从有限时间区间推广到了无限时间区间。在[4]中，Chen给出并证明了带停时的倒向随机微分方程解的存在唯一性定理。我们认为带停时的倒向重随机微分方程也能给出一类拟线性随机偏微分方程的解的概率表示。仿照Chen和Wang的方法，我们给出了带停时的倒向重随机微分方程的解的存在唯一性定理，并证明了连续依赖性和连续收敛性定理。
本文的结构如下：在第二章中，我们列出了在这篇文章中用到的几个重要的定理。在第三章中，我们给出了所要研究问题的基本设定和主要假设。第四章，我们首先在第一节中提出并证明了文章的主要结果-存在唯一性定理，然后在第二节中讨论了解的连续依赖性和收敛性定理。
In this paper, we give a sufficient condition on the coefficients of a class of backward doubly stochastic differential equations with stopping time (BDSDEs) under which the BDSDEs with stopping time have a unique solution for any given square integrable terminal values. We also discuss the continuous dependence theorem and convergence theorem for this class of equations.
Since the backward stochastic differential equations (BSDEs in short) was introduced by Pardoux and Peng in 1990, the theory of BSDEs has been developed and widely used by many researchers. The main result is, for any givenξ∈L~2(Ω, F_T, P; R~k) , the following BSDEhas a unique solution pair (y_t,z_t) in the interval [0, T] under some condition. This class of equations provide a useful framework for formulating many problems in mathematical finance, and they are also useful for problems in stochastic control, stochastic differential game, and probabilistic formula for the solutions of quasi-linear partial differential equations.
After Pardoux and Peng (1990) introduced the theory of BSDEs, Pardoux and Peng introduced a new class of backward stochastic differential equations-backward "doubly" stochastic differential equations and also showed the existence and uniqueness theorem of the solution of BDSDEs in a finite time interval. These research allow us to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDEs. In [3], Chen and Wang extended BSDEs from finite time interval [0,T] to infinite time interval [0,∞]. Chen also proved the existence and uniqueness theorem for BSDEs with stopping time in [4]. BDSDEs with stopping time are also very interesting to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations. Here following the approach of Chen and Wang, we present the existence and uniqueness theorem for BDSDEs with stopping time and proved the continuous dependence theorem and convergence theorem.
This paper is organized as follows. In chapter 2, we show some important theorems we used in our paper. In chapter 3, we present the setting of the problem and the main assumptions. In chapter 4, first we present and prove the main result-the existence and uniqueness theorem. Then we discuss the continuous dependence theorem and convergence theorem.
引文
[1] Pardoux E.& Peng, S., 1990. Adapted solution of a backward stochastic differential equation. Systems Control Letters 14, 55-61.
[2] Pardoux E.& Peng, S., 1994. Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDEs. Probab. Theory Relat. Fields 98, 209-227.
[3] Chen, Z.& Wang, B., 2000. Infinite time interval BSDEs and the convergence of g-martingales. J. Austral. Math. Soc. (Series A) 69, 187-211.
[4] Chen, Z., 1998. Existence and uniqueness for BSDE with stopping time. Chinese Sci Bull 43 (2), 96-99.
[5] Darling, R.W.R. & Pardoux E., 1997. Backward stochastic differential equations with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25,1135-1159.
[6] El Karoui, N. & Peng, S.& Quenez, M.C., 1997. Backward stochastic differential equations in finance. Math. Finance. 7 (1), 1-71.
[7] Ma J.& Protter P.& Yong J., 1994. Solving forward-backward stochastic differential equations explicitly-a four step scheme. Prabab. Theory Relat. Fields 98, 339-359.
[8] Ma J.& Yong J., 1999. Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. Vol. 1702, Springer-Verlag, Berlin.
[9] Naulart, D. & Pardoux, E., 1998. Stochastic calculus with anticipating integrands.Prabab. Theory Relat. Fields 78, 535-581.
[10] Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics. 37, 61-74.
[11] Peng, S., 1997. Backward stochastic differential equations and related g-expectation. In:El Karoui, N., Mazliak, L. (Eds.), Backward Stochastic Differential Equations. Longman,Harlow, pp. 141-159.
[12] Peng, S.& Shi, Y., 2000. Infinite horizon forward-backward stochastic differential equations.Stoch. Proc. Appl. 85, 75-92.
[13] Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics 37 (1991), 61-74.
[14] HU, Y. & Peng, S., 1995. Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields 103 (1995), 273-283.
[15] Nualart, D.& Pardoux,E., 1988. Stochastic evolution equations. J.Sov.Math. 16, 1233-1277.
[16] (?)ksendal, B., 1995. Stochastic Differential Equation. Springer-Verlag, 43-60.
[17] Artzner, P.& Delbaen, F.& Eber, J.M. & Heath, D., 1999. Coherent measures of risk. Math. Finance, 9, 203-228.
[18] Bismut, J.-M., 1978. An introductory approach to duality in optimal stochastic control. SIAM Rev., 20 (1), 62-78.
[19] Chen, Z.& Epstein, L., 2002. Ambiguity, Risk and Asset Returns in Continuous Time. Econometrica, 70 (4), 1403-1443.
[20] Coquet, F.& Y. Hu J. Memin & Peng, S., 2002. Filtration-consistant nonlinear expectationsand related g-expectations. Probab. Theory Relat. Fields, 123, 1-27.
[21] Matoussi, A.& Scheutzow, M.,2002. Stochastic PDES driven by nonlinear noise and backward doubly SDEs driven by nonlinear noise and backwardd oubly SDEs [J].J. Theorey Probab.. 15, 1-39.
[22] Guichang, Z., 2006. Discretization of backward semilinear stochastic evolution equations [J]. Proba Theory and Relat.Fields, Republished.
[23] Mao, X.,1995. Adapted solutions of backward stochastic differential equations with Non-Lipshictz coefficients [J]. Stochastic Process and their Applications, 58, 281-292.
[24] Peng, S.& Shi, Y., 1999. Infinite Horizon Boundary Value Problems and Applications. Journal of differential equations 155, 405-422.
[25] Peng, S.& Wu, Z., 1999. Fully coupled forward-backward stochastic differential equationsand applications to optimal control. SIAM J. Control Optim, 37 (1999), 825-843.
[26] Yong, J., 1997. Finding adapted solutions of forward-backward stochastic differential equations-Method of continuation. Probab. Theory Relat. Fields 107 (1997), 537-572.
[2] Pardoux E.& Peng, S., 1994. Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDEs. Probab. Theory Relat. Fields 98, 209-227.
[3] Chen, Z.& Wang, B., 2000. Infinite time interval BSDEs and the convergence of g-martingales. J. Austral. Math. Soc. (Series A) 69, 187-211.
[4] Chen, Z., 1998. Existence and uniqueness for BSDE with stopping time. Chinese Sci Bull 43 (2), 96-99.
[5] Darling, R.W.R. & Pardoux E., 1997. Backward stochastic differential equations with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25,1135-1159.
[6] El Karoui, N. & Peng, S.& Quenez, M.C., 1997. Backward stochastic differential equations in finance. Math. Finance. 7 (1), 1-71.
[7] Ma J.& Protter P.& Yong J., 1994. Solving forward-backward stochastic differential equations explicitly-a four step scheme. Prabab. Theory Relat. Fields 98, 339-359.
[8] Ma J.& Yong J., 1999. Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. Vol. 1702, Springer-Verlag, Berlin.
[9] Naulart, D. & Pardoux, E., 1998. Stochastic calculus with anticipating integrands.Prabab. Theory Relat. Fields 78, 535-581.
[10] Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics. 37, 61-74.
[11] Peng, S., 1997. Backward stochastic differential equations and related g-expectation. In:El Karoui, N., Mazliak, L. (Eds.), Backward Stochastic Differential Equations. Longman,Harlow, pp. 141-159.
[12] Peng, S.& Shi, Y., 2000. Infinite horizon forward-backward stochastic differential equations.Stoch. Proc. Appl. 85, 75-92.
[13] Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics 37 (1991), 61-74.
[14] HU, Y. & Peng, S., 1995. Solution of forward-backward stochastic differential equations. Probab. Theory Relat. Fields 103 (1995), 273-283.
[15] Nualart, D.& Pardoux,E., 1988. Stochastic evolution equations. J.Sov.Math. 16, 1233-1277.
[16] (?)ksendal, B., 1995. Stochastic Differential Equation. Springer-Verlag, 43-60.
[17] Artzner, P.& Delbaen, F.& Eber, J.M. & Heath, D., 1999. Coherent measures of risk. Math. Finance, 9, 203-228.
[18] Bismut, J.-M., 1978. An introductory approach to duality in optimal stochastic control. SIAM Rev., 20 (1), 62-78.
[19] Chen, Z.& Epstein, L., 2002. Ambiguity, Risk and Asset Returns in Continuous Time. Econometrica, 70 (4), 1403-1443.
[20] Coquet, F.& Y. Hu J. Memin & Peng, S., 2002. Filtration-consistant nonlinear expectationsand related g-expectations. Probab. Theory Relat. Fields, 123, 1-27.
[21] Matoussi, A.& Scheutzow, M.,2002. Stochastic PDES driven by nonlinear noise and backward doubly SDEs driven by nonlinear noise and backwardd oubly SDEs [J].J. Theorey Probab.. 15, 1-39.
[22] Guichang, Z., 2006. Discretization of backward semilinear stochastic evolution equations [J]. Proba Theory and Relat.Fields, Republished.
[23] Mao, X.,1995. Adapted solutions of backward stochastic differential equations with Non-Lipshictz coefficients [J]. Stochastic Process and their Applications, 58, 281-292.
[24] Peng, S.& Shi, Y., 1999. Infinite Horizon Boundary Value Problems and Applications. Journal of differential equations 155, 405-422.
[25] Peng, S.& Wu, Z., 1999. Fully coupled forward-backward stochastic differential equationsand applications to optimal control. SIAM J. Control Optim, 37 (1999), 825-843.
[26] Yong, J., 1997. Finding adapted solutions of forward-backward stochastic differential equations-Method of continuation. Probab. Theory Relat. Fields 107 (1997), 537-572.
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