无穷区间多维反射倒向随机微分方程和比较定理
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摘要
本文研究的是无穷区间多维反射倒向随机微分方程解的存在唯一性，解对参数的连续依赖性以及比较定理。
众所周知，倒向随机微分方程(BSDE)是一个新兴的研究方向，它的出现为研究金融数学，随机最优控制以及偏微分方程等问题提供了有利的工具。如下的非线性BSDE：
-dY(t)=f(t，Y(t)，Z(t)dt-Z(t)dB_t，Y_T=ξ是由Pardoux和Peng于1990年在[1]中首先介绍的，后来Peng于1992年在[2]中证明了一维BSDE的比较定理，彭实戈教授的学生周海滨于1999年在[3]中证明了一类多维BSDE的比较定理，证明方法是构造了一个特殊的函数，这个函数是Buckdahn和Peng于1999年在[4]中首次介绍的。El.Karouietal在[5]中研究了带一个障碍的一维反射BSDE，给出了一维情况下解的存在唯一性定理和比较定理，同时还在Markov框架下研究了一维反射BSDE与非线性抛物性偏微分方程的联系。Hamadene，Lepeltier，Wu把这种一维反射BSDE的区间扩展到了无穷区间上，而肖华则在2005年的硕士毕业论文中把一维反射BSDE推广到了多维的情况。
在这些基础之上，我们现在很自然的提出，如何建立无穷区间上多维反射BSDE的框架，建立之后，无穷区间多维反射BSDE的解是否存在唯一，解对参数是否具有连续依赖性以及解的比较定理是否成立。
本文共分三章。
第一章：引言，叙述前人所作的工作以及问题的由来。
第二章：在[6]中对一维情况的证明以及[7]中对于反射BSDE从一维到多维的推广的基础上，我们提出了如下的无穷区间多维反射BSDE模型：
首先假定：
(H2．1)ξ∈L_n~2。
(H2．2)f：Ω×[0，∞]×R~n×R~(n×d)→R~n，f(·，y，z)是循序可测的并满足：
E(integral from n=0 to∞|f_j(s，0，0)|ds)~2＜∞，(?)j=1，2，…，n。
(H2．3)存在两个正的确定的函数u_1(t)，u_2(t)，使得：
|f(t，y，z)-f(t，y′，z′)|≤u_1(t)|y-y′|+u_2(t)|z-z′|其中t≥0，y，y′∈R~n，z，z′∈R~(n×d)且有integral from n=0 to∞u_1(t)dt＜∞，integral from n=0 to∞u_2~2(t)dt＜∞。
下面再给出一个n维障碍{S(t)，t≥0}∈R~n满足：
(H2．4)S(t)是连续的循序可测过程且满足：
E[(?)(S~+(t))~2]＜∞，lim (?) S(t)≤ξ．我们称(f，ξ，S)为一组标准参数，若它满足(H2．1)-(H2．4)，称{Y(t)，Z(t)，K(t)，t≥0}∈R~n×R~(n×d)×R~(n~+)是无穷区间n维RBSDE的一组解，若它满足：
(H2．5)Y(t)∈S_n~2，Z(t)∈H_n~2，K(∞)∈L_n~2。
(H2．6)Y(t)=ξ+integral from n=t to∞f(s，Y(s)，Z(s))ds+K(∞)-K(t)-integral from n=t to∞Z(s)dB_s。
(H2．7)Y(t)≥S(t)，t≥0。
(H2．8)K(t)是连续的增过程，K(0)=0且integral from n=0 to∞(Y(t)-S(t))dK(t)=0。
这里Y(t)是一个R~n向量，它的第j个分量是Y_j(t)。
多维模型同一维模型的区别主要体现在(H2．7)和(H2．8)上，意味着仅当Y_j碰到障碍S_j时，用一个最小的推动力K_j将Y_j向上推动，使之在障碍S_j上面运动。以[6]与[7]对反射BSDE的证明为基础，我们证明了无穷区间多维反射BSDE解的存在唯一性，即
定理2．1：当(f，ξ，S)满足(H2．1)-(H2．4)的条件时，无穷区间多维RBSDE(f，ξ，S)存在着一组解(Y，Z，K)满足条件(H2．5)-(H2．8)且至多有一组循序可测的解。
有了解的存在唯一性，在这一章的最后，我们还证明了解对参数是具有连续依赖性的。
第三章：为了说明无穷区间多维反射BSDE的比较定理，我们将两个R~n中向量的比较定义为：
α~1≥α~2(?)α_j~1≥α_j~2，j=1，2，…，n。α~1，α~2∈R~n。由此得出了无穷区间多维BSDE的比较定理：
定理3．1：设(f~i，ξ~i，s~i)和(Y~i，Z~i，K~i)，i=1，2，分别满足条件(H2．1)-(H2．8)，并且
(ⅰ)ξ~1≤ξ~2；
(ⅱ)f_j~1(t，y~1，z~1)≤f_j~2(t，y~2，z~2)，其中y_j~1=y_j~2，z_j~1=z_j~2，y_l~1≤y_l~2，l≠j；
(ⅲ)S~1≤S~2。
则有Y~1≤Y~2。
下面，考虑定理3．1中的条件(ⅱ)能否换成更弱的条件：
(ⅱ′)f_j~1(t，y，z~1)≤f_j~2(t，y，z~2)，z_j~1=z_j~2我们举了一个满足条件(ⅱ′)但不满足条件(ⅱ)的例子，通过这个例子证明了比较定理不一定成立。
In this paper,we study the infinite horizon multi-dimensional reflected backward stochastic differential equations (BSDE in short). The existence and uniqueness result of the solution for this kind of equation was proved, and we also give one kind of multi-dimensional comparison theorem for the reflected BSDE .
It is well known that BSDE has become a field of increasing activity.It is becoming an important tool in study of financial mathematics,stochastic optimal control problems and partial differential equation. The following nonlinear BSDE
was first introduced by Pardoux and Peng in1990(see[1]).After that, Peng prove the comparison theorem for one-dimensional BSDE in 1992(see[2]).Zhou proved one kind of comparison theorem for multi-dimensional BSDE in1999(see[3]),and then use the comparison theorem as the tool to prove one existence result for multi-dimensional BSDE where the coefficient is continuous and has the linear growth.El.Karoui er al studied one-dimensional reflected BSDE with one barrier in 1997(see[5]),and then proved the existence a nd uniqueness result and comparison theorem of the solution for this kind of equation.
we divide this paper into three chapters.
The first chapter is an introduction.
In Chapter 2,we give the following model for multi-dimensional reflected BSDE. We first give the comparison definition for two vectors in R~n: and then assume:
(H2.3) There are two positive functions u_1(t),u_2(t), satisfying
where t ≥ 0, y,y' ∈ R~n, z, z' ∈ H~(n×d) and U_1(t)dt < ∞, u_2~2(t)dt < ∞. and we give one n-dimensional obstacle{5(f),t ≥ 0} ∈ R~n satisfying (H2.4) S(t) is a continuous progressively measurable R~n-valued process satisfying
(f, ξ,S) is called one group of standard parameter for n-dimensional reflected BSDE, if it is satisfies (H2.1)-(H2.4).
And then we call {(Y(t),Z(t),K(t)),t ≥ 0} to be the solution for infinite horizon n-dimensional reflected BSDE if it satisfies
(H2.8) K(t) is continuous and increase process satisfying K(0) = 0 and (Y(t)-S(t))dK(t) = 0.
Theorem 2.1: We assume (f,ξ,S) satisfies (H2.1)-(H2.4), then there exists a group of solution (Y,Z,K) for n-dimensional reflected BSDE satisfying (H2.5)-(H2.8) , which is also unique.
After we proved the existence and uniqueness of the solution, we have also proved the solution is continuous about the parameter.
In Chapter 3,we get Theorem 3.1: Let (f~1 ξ~1, S~1) and(f~2, ξ~2, S~2) be two standard parameters of the n-dimensional reflected BSDE satisfying(H2.1)-(H2.4), and suppose in addition the following
then Y~1 ≤ Y~2 .
Then we think about whether it is possible to change to the weaker assumption:
From the counterexample 3.1, we know that the comparison theorem does not hold under the assumption (ii').
众所周知，倒向随机微分方程(BSDE)是一个新兴的研究方向，它的出现为研究金融数学，随机最优控制以及偏微分方程等问题提供了有利的工具。如下的非线性BSDE：
-dY(t)=f(t，Y(t)，Z(t)dt-Z(t)dB_t，Y_T=ξ是由Pardoux和Peng于1990年在[1]中首先介绍的，后来Peng于1992年在[2]中证明了一维BSDE的比较定理，彭实戈教授的学生周海滨于1999年在[3]中证明了一类多维BSDE的比较定理，证明方法是构造了一个特殊的函数，这个函数是Buckdahn和Peng于1999年在[4]中首次介绍的。El.Karouietal在[5]中研究了带一个障碍的一维反射BSDE，给出了一维情况下解的存在唯一性定理和比较定理，同时还在Markov框架下研究了一维反射BSDE与非线性抛物性偏微分方程的联系。Hamadene，Lepeltier，Wu把这种一维反射BSDE的区间扩展到了无穷区间上，而肖华则在2005年的硕士毕业论文中把一维反射BSDE推广到了多维的情况。
在这些基础之上，我们现在很自然的提出，如何建立无穷区间上多维反射BSDE的框架，建立之后，无穷区间多维反射BSDE的解是否存在唯一，解对参数是否具有连续依赖性以及解的比较定理是否成立。
本文共分三章。
第一章：引言，叙述前人所作的工作以及问题的由来。
第二章：在[6]中对一维情况的证明以及[7]中对于反射BSDE从一维到多维的推广的基础上，我们提出了如下的无穷区间多维反射BSDE模型：
首先假定：
(H2．1)ξ∈L_n~2。
(H2．2)f：Ω×[0，∞]×R~n×R~(n×d)→R~n，f(·，y，z)是循序可测的并满足：
E(integral from n=0 to∞|f_j(s，0，0)|ds)~2＜∞，(?)j=1，2，…，n。
(H2．3)存在两个正的确定的函数u_1(t)，u_2(t)，使得：
|f(t，y，z)-f(t，y′，z′)|≤u_1(t)|y-y′|+u_2(t)|z-z′|其中t≥0，y，y′∈R~n，z，z′∈R~(n×d)且有integral from n=0 to∞u_1(t)dt＜∞，integral from n=0 to∞u_2~2(t)dt＜∞。
下面再给出一个n维障碍{S(t)，t≥0}∈R~n满足：
(H2．4)S(t)是连续的循序可测过程且满足：
E[(?)(S~+(t))~2]＜∞，lim (?) S(t)≤ξ．我们称(f，ξ，S)为一组标准参数，若它满足(H2．1)-(H2．4)，称{Y(t)，Z(t)，K(t)，t≥0}∈R~n×R~(n×d)×R~(n~+)是无穷区间n维RBSDE的一组解，若它满足：
(H2．5)Y(t)∈S_n~2，Z(t)∈H_n~2，K(∞)∈L_n~2。
(H2．6)Y(t)=ξ+integral from n=t to∞f(s，Y(s)，Z(s))ds+K(∞)-K(t)-integral from n=t to∞Z(s)dB_s。
(H2．7)Y(t)≥S(t)，t≥0。
(H2．8)K(t)是连续的增过程，K(0)=0且integral from n=0 to∞(Y(t)-S(t))dK(t)=0。
这里Y(t)是一个R~n向量，它的第j个分量是Y_j(t)。
多维模型同一维模型的区别主要体现在(H2．7)和(H2．8)上，意味着仅当Y_j碰到障碍S_j时，用一个最小的推动力K_j将Y_j向上推动，使之在障碍S_j上面运动。以[6]与[7]对反射BSDE的证明为基础，我们证明了无穷区间多维反射BSDE解的存在唯一性，即
定理2．1：当(f，ξ，S)满足(H2．1)-(H2．4)的条件时，无穷区间多维RBSDE(f，ξ，S)存在着一组解(Y，Z，K)满足条件(H2．5)-(H2．8)且至多有一组循序可测的解。
有了解的存在唯一性，在这一章的最后，我们还证明了解对参数是具有连续依赖性的。
第三章：为了说明无穷区间多维反射BSDE的比较定理，我们将两个R~n中向量的比较定义为：
α~1≥α~2(?)α_j~1≥α_j~2，j=1，2，…，n。α~1，α~2∈R~n。由此得出了无穷区间多维BSDE的比较定理：
定理3．1：设(f~i，ξ~i，s~i)和(Y~i，Z~i，K~i)，i=1，2，分别满足条件(H2．1)-(H2．8)，并且
(ⅰ)ξ~1≤ξ~2；
(ⅱ)f_j~1(t，y~1，z~1)≤f_j~2(t，y~2，z~2)，其中y_j~1=y_j~2，z_j~1=z_j~2，y_l~1≤y_l~2，l≠j；
(ⅲ)S~1≤S~2。
则有Y~1≤Y~2。
下面，考虑定理3．1中的条件(ⅱ)能否换成更弱的条件：
(ⅱ′)f_j~1(t，y，z~1)≤f_j~2(t，y，z~2)，z_j~1=z_j~2我们举了一个满足条件(ⅱ′)但不满足条件(ⅱ)的例子，通过这个例子证明了比较定理不一定成立。
In this paper,we study the infinite horizon multi-dimensional reflected backward stochastic differential equations (BSDE in short). The existence and uniqueness result of the solution for this kind of equation was proved, and we also give one kind of multi-dimensional comparison theorem for the reflected BSDE .
It is well known that BSDE has become a field of increasing activity.It is becoming an important tool in study of financial mathematics,stochastic optimal control problems and partial differential equation. The following nonlinear BSDE
was first introduced by Pardoux and Peng in1990(see[1]).After that, Peng prove the comparison theorem for one-dimensional BSDE in 1992(see[2]).Zhou proved one kind of comparison theorem for multi-dimensional BSDE in1999(see[3]),and then use the comparison theorem as the tool to prove one existence result for multi-dimensional BSDE where the coefficient is continuous and has the linear growth.El.Karoui er al studied one-dimensional reflected BSDE with one barrier in 1997(see[5]),and then proved the existence a nd uniqueness result and comparison theorem of the solution for this kind of equation.
we divide this paper into three chapters.
The first chapter is an introduction.
In Chapter 2,we give the following model for multi-dimensional reflected BSDE. We first give the comparison definition for two vectors in R~n: and then assume:
(H2.3) There are two positive functions u_1(t),u_2(t), satisfying
where t ≥ 0, y,y' ∈ R~n, z, z' ∈ H~(n×d) and U_1(t)dt < ∞, u_2~2(t)dt < ∞. and we give one n-dimensional obstacle{5(f),t ≥ 0} ∈ R~n satisfying (H2.4) S(t) is a continuous progressively measurable R~n-valued process satisfying
(f, ξ,S) is called one group of standard parameter for n-dimensional reflected BSDE, if it is satisfies (H2.1)-(H2.4).
And then we call {(Y(t),Z(t),K(t)),t ≥ 0} to be the solution for infinite horizon n-dimensional reflected BSDE if it satisfies
(H2.8) K(t) is continuous and increase process satisfying K(0) = 0 and (Y(t)-S(t))dK(t) = 0.
Theorem 2.1: We assume (f,ξ,S) satisfies (H2.1)-(H2.4), then there exists a group of solution (Y,Z,K) for n-dimensional reflected BSDE satisfying (H2.5)-(H2.8) , which is also unique.
After we proved the existence and uniqueness of the solution, we have also proved the solution is continuous about the parameter.
In Chapter 3,we get Theorem 3.1: Let (f~1 ξ~1, S~1) and(f~2, ξ~2, S~2) be two standard parameters of the n-dimensional reflected BSDE satisfying(H2.1)-(H2.4), and suppose in addition the following
then Y~1 ≤ Y~2 .
Then we think about whether it is possible to change to the weaker assumption:
From the counterexample 3.1, we know that the comparison theorem does not hold under the assumption (ii').
引文
[1] E. Pardoux, S. peng, Adapted solutions of a backward stochastic differential equation, Systems and Control Letters, 14, 55-61(1990).
[2] S. Peng, A Generalized Dynamic Programming Principle and Hamilton-Jacobi-Bellman Equation, Stochastic and Stochastic Reports, 38,119-134(1992).
[3] 周海滨，多维倒向随机微分方程的比较定理及应用，山东大学硕士论文，1999
[4] R. Budkdahn, S. Peng, Stationary backward stochastic differential equations and associated partial differential equations, Probability Theory and related fields, 115, 383-399 (1999).
[5] N. El-Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.. C. Quenez, Reflected solutions of backward SDE's and related obstacle problems for PDE's, Ann. Probab. 25(2), 702-737 (1997).
[6] S. Hamadene, J-P. Lepeltier, Z Wu, Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems, Probability and Mathematical, 19, 211-234, 1999
[7] 肖华，多维反射倒向随机微分方程和比较定理，山东大学硕士论文，2005
[8] D. Duffie, L. Epstein, Stochastic differential utility, Econometrica, 60, 353-394 (1992).
[9] N. EI-Karoui, S. Peng, M. C. Quenez, Backward stochastic differential equation in finance, Math Finance, 7, 1-71 (1997).
[10] S. Hamadene, J. P. Lepeltier, Backward equations,stochastic control and zero-sum stochastic differential games, Stochastics and stochastic reports, 54, 221-231 (1995).
[11] J. Cvitanic, I. Karatzas, Backward SDE's with reflection and Dlynkin games, Ann. Probab, 24(4), 2024-2056 (1996).
[12] S. Hamadene, J. P. Lepeltier, Reflected BSDE and mixed games, Stochastic Process Appl, 85, 177-188 (2000).
[13] C.Gei, R.Manthey, Comarison theorems for stochastic differential equations in finite and in finite dimensions, Stochastic Process Appl, 53, 23-35 (1994).
[14] J.P.Lepeltier, J.San.Martin, Backward stochastic differential equations with continuous coefficient, Statistics and Probability letters, 32, 425-430 (1997).
[15] Mel'nikov.A.V., Onsolutions of stochastic differential equation with driving semi-martingales, Prroc European Young Statisticians Meeting, Leuven 120-124 (1983).
[16] S.Assing, R.Manthey, The behavior of solutions of stochastic differential inequalities, Probability Theory and Related Fields,103,493-514 (1995).
[17] X.Ding, R.Wu, A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales , to appear in Stochastic Process and their Applications.
[18] M.T.Barlow, E.Perkins, One dimensional stochastic differential equations involving a singular increasing process, Stochastic, 12, 229-249 (1984).
[19] M.Alario-Nazaret, Jeux de Dynkin, These de Doctorat en mathematiques, Unibersite de Franche-Comte,Fance, 1982.
[20] A.Bensoussanet J.L.Lions, Applications des inequations variationnelles en controle stochastique, Dunod, Paris 1979.
[21] Z.Chen, Existence and uniqueness for BSDE's with stopping time, Chinese Science Bulletin 43 (1998),pp.96-99.
[2] S. Peng, A Generalized Dynamic Programming Principle and Hamilton-Jacobi-Bellman Equation, Stochastic and Stochastic Reports, 38,119-134(1992).
[3] 周海滨，多维倒向随机微分方程的比较定理及应用，山东大学硕士论文，1999
[4] R. Budkdahn, S. Peng, Stationary backward stochastic differential equations and associated partial differential equations, Probability Theory and related fields, 115, 383-399 (1999).
[5] N. El-Karoui, C. Kapoudjian, E. Pardoux, S. Peng, M.. C. Quenez, Reflected solutions of backward SDE's and related obstacle problems for PDE's, Ann. Probab. 25(2), 702-737 (1997).
[6] S. Hamadene, J-P. Lepeltier, Z Wu, Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems, Probability and Mathematical, 19, 211-234, 1999
[7] 肖华，多维反射倒向随机微分方程和比较定理，山东大学硕士论文，2005
[8] D. Duffie, L. Epstein, Stochastic differential utility, Econometrica, 60, 353-394 (1992).
[9] N. EI-Karoui, S. Peng, M. C. Quenez, Backward stochastic differential equation in finance, Math Finance, 7, 1-71 (1997).
[10] S. Hamadene, J. P. Lepeltier, Backward equations,stochastic control and zero-sum stochastic differential games, Stochastics and stochastic reports, 54, 221-231 (1995).
[11] J. Cvitanic, I. Karatzas, Backward SDE's with reflection and Dlynkin games, Ann. Probab, 24(4), 2024-2056 (1996).
[12] S. Hamadene, J. P. Lepeltier, Reflected BSDE and mixed games, Stochastic Process Appl, 85, 177-188 (2000).
[13] C.Gei, R.Manthey, Comarison theorems for stochastic differential equations in finite and in finite dimensions, Stochastic Process Appl, 53, 23-35 (1994).
[14] J.P.Lepeltier, J.San.Martin, Backward stochastic differential equations with continuous coefficient, Statistics and Probability letters, 32, 425-430 (1997).
[15] Mel'nikov.A.V., Onsolutions of stochastic differential equation with driving semi-martingales, Prroc European Young Statisticians Meeting, Leuven 120-124 (1983).
[16] S.Assing, R.Manthey, The behavior of solutions of stochastic differential inequalities, Probability Theory and Related Fields,103,493-514 (1995).
[17] X.Ding, R.Wu, A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales , to appear in Stochastic Process and their Applications.
[18] M.T.Barlow, E.Perkins, One dimensional stochastic differential equations involving a singular increasing process, Stochastic, 12, 229-249 (1984).
[19] M.Alario-Nazaret, Jeux de Dynkin, These de Doctorat en mathematiques, Unibersite de Franche-Comte,Fance, 1982.
[20] A.Bensoussanet J.L.Lions, Applications des inequations variationnelles en controle stochastique, Dunod, Paris 1979.
[21] Z.Chen, Existence and uniqueness for BSDE's with stopping time, Chinese Science Bulletin 43 (1998),pp.96-99.