关于随机微分方程初值解的存在唯一性定理的推广
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摘要
随机微分方程是近几十年兴起的热门的数学学科,在许多领域有着广泛的应用。是数学中的一个非常活跃且引人瞩目的领域,国内和国际上许多著名的数学家都投入到这一领域进行研究并获得了一些辉煌的成果,使得人们对于自然界无处不在的随机现象有了更加深刻的理解。随机微分方程是介于微分方程与概率论之间的边缘分支,它是两个数学分支互相渗透的结果。它对数学领域中的许多分支起着有效的联结作用。在随机微分方程的定性理论研究中,随机微分方程初值解的存在唯一性是非常重要的课题。本文的主要工作是推广了随机微分方程和随机泛函微分方程的初值解的存在唯一性定理,减弱经典的Lipschitz条件(一致Lipschitz条件)和线性增长条件,仍保证随机微分方程和随机泛函微分方程的初值解的存在唯一性,并给出具体例子来说明推广后的定理适用于更广泛的方程。并给出了随机微分方程与随机泛函微分方程的近似解与精确解之间的误差估计表达式,同时本文也给出了随机微分方程和随机泛函微分方程初值解的新的指数估值。最后,总结了论文的创新点,提出了论文的改进方向,并给出了研究中所参考的主要文献。
The theory of stochastic differential equations has a rapidly developing life of itsown as a fascinating research field from those decades. It has a wide range of applica-tions outside mathematics, there are many fruitful connections to other mathematicaldisciplines. There are many famous mathematicians have interested in studying thisfield , and had made a number of results. And then so many people seem to placemuch more attention on stochastic phenomenons in nature. Stochastic differentialequations is the edge branches of differential equations and probability theory, andplay an effective role to associate many mathematic branches. In the process of qual-itative theory developed, the existence and uniqueness of solutions to SDE is veryimportant. This paper is devoted to improve the existence and uniqueness of solutionsto SDE and SFDE. Weak the general Lipschitz condition (uniformly Lipschitz condi-tion) and linear growth condition, and establish an improved existence and uniquenesstheorem about the stochastic differential equation and stochastic functional differen-tial equation with random initial value. With the promotion, the theory extends to bemore feasible for a broad class of equations. We give the estimate for the error of theapproximate solutions and the real solutions. The new exponential estimates of thesolution are also to be given. Finally, this paper sums up the innovations and gives thedirection to improve and the study in references on the main results.
The theory of stochastic differential equations has a rapidly developing life of itsown as a fascinating research field from those decades. It has a wide range of applica-tions outside mathematics, there are many fruitful connections to other mathematicaldisciplines. There are many famous mathematicians have interested in studying thisfield , and had made a number of results. And then so many people seem to placemuch more attention on stochastic phenomenons in nature. Stochastic differentialequations is the edge branches of differential equations and probability theory, andplay an effective role to associate many mathematic branches. In the process of qual-itative theory developed, the existence and uniqueness of solutions to SDE is veryimportant. This paper is devoted to improve the existence and uniqueness of solutionsto SDE and SFDE. Weak the general Lipschitz condition (uniformly Lipschitz condi-tion) and linear growth condition, and establish an improved existence and uniquenesstheorem about the stochastic differential equation and stochastic functional differen-tial equation with random initial value. With the promotion, the theory extends to bemore feasible for a broad class of equations. We give the estimate for the error of theapproximate solutions and the real solutions. The new exponential estimates of thesolution are also to be given. Finally, this paper sums up the innovations and gives thedirection to improve and the study in references on the main results.
引文
1 X. Mao. Stochastic Differential Equations and Their Applications. HorwoodPublication, Chichester, 1997:35～162
2 T.C. Gard. Introduction to stochastic differential Equations. Marcel Dekker, NewYork, 1988:52～89
3 A. Friedman. Stochastic Differential Equations and Their Applications. Aca-demic Press, San Diego, 1976:39～120
4 Bernt, Ksendal. Stochastic Differential Equations. 6th Edition. Springer-VerlagBerlin, 2005:63～96
5东北师范大学微分方程教研室.常微分方程(第二版).高等教育出版社,2005:109～114
6尤秉礼.常微分方程补充教程.人民教育出版社, 1981:23～98
7 J. Hale, S.M.V. Lunel. Introduction to Functions Differential Equatons. NewYork:Springer, 1993:13～69
8 J. Littner. The Valuation of Risk Assets and the Selection of Risk Investments inStock Portfolios and Capital Budgets. Review of Economics and Statistic. 1965,47:13～37
9 S. Ross. The Arbitrage Theory of Capital Asset Pricing. Journal of EconomicTheory. 1976, 13(3):341～360
10 RC. Merton. The Theory of Rational Option Pricing. Bell Journal of Economics& Manag. 1973, 49:143～183
11 F. Wei, K. Wang. The Existence and Uniqueness of the Solution for StochasticFunctional Differential Equations with Infinite Delay. Journal of MathematicalAnalysis and Applications. 2007, 331(1):516～531
12 I. Gikhman, A. Skorokhood. Stochastc Differential Equations. Springer-Verlag,New York, 1974:35～87
13 L. Arnold. Stochastic Differential Equations Theory and Applications. Wily, NewYork, 1974:39～86
14 L. Arnold. Random Dynamical Systems. Spring-Verlag, 1998:57～110
15 EI Karoui N, S. Peng, M C. Quence. Backward stochastic differential equationin finance. Math Finance. 1997, 7(1):1～13
16 X. Mao. Stotistics Functional Differential Equations. Springer, 1976:36～81
17 T.C. Gard. Pathwise Uniqueness Solutions of Systems Integrals and Equations.Stochastic Process and Application. 1978, 6:253～260
18 X. Mao. Existence and Uniqueness of the Solutions of Delay Differential Equa-tions. Stochastic Analysis and Applications. 1989, 7(1):59～74
19 T.C. Gard. A General Uniqueness for Theory for Stochastic Differential Equa-tions. SIAM Journal on Control and Optimization. 1976, 14:445～457
20周少波,薛明皋.无限时滞中立型随机泛函微分方程解的存在唯一性.应用数学. 2008, 28(1):75～83
21 G. Cao. Existence and Pathwise Uniqueness of Solutions to Stochastic Dif-ferential Equations in Banach Space. Mathematica Applicata(China). 2006,19(1):75～79
22冉启康.一类非Lipschitz条件的BSDE解的存在唯一性.工程数学学报.2006, 23(2):286～292
23曹桂兰,何凯.由无穷个Brown单驱动的随机微分方程解的存在唯一性.数学物理学报. 2006, 26(6):813～823
24孙晓君.非Lipschitz条件下多维随机微分方程强解的存在唯一性.工程数学学报. 1998, 01:121～124
25聂赞坎.平面中随机微分方程解的存在唯一性.高校应用数学学报A辑(中文版). 1987, 2(2):222～227
26汪家义,许明浩.随机微分-积分方程解的存在唯一性.数学杂志. 1999,03:333～338
27史树中.凸分析.上海科学技术出版社, 1990:93～123
28 J. Martin Lindsay, Adam G. Skalski. On Quantum Stochastic Differen-tial Equations. Journal of Mathematical Analysis and Applications. 2007,330(2):1093～1114
29 F. Wang. Ito? type Measure-valued Stochastic Differential Equations. Journal ofMathematical Analysis and Applications. 2007, 329(2):1102～1117
30 H. Schurz. Existence and Uniqueness of Solutions of Semilinear Stochas-tic Infinite-Dimensional Differential Systems with H-regular Noise. Journal ofMathematical Analysis and Applications. 2006, 332(1):334～345
31 M. Rei, M. Riedle, O. van Gaans. Delay Differential Equations Driven By LévyProcesses: Stationarity and Feller Properties. Stochastic Processes and their Ap-plications. 2006, 11610:1409～1432
32 Y. Shen, L. Qi, X. Mao. The Improved LaSalle-type Theorems for StochasticFunctional Differential Equations. Journal of Mathematical Analysis and Appli-cations. 2006, 318(1):134～154
33 D. Nualart, P.A. Vuillermot. Variational Solutions for Partial Differential Equa-tions Driven By a Fractional Noise. Journal of Functional Analysis. 2006,232(2):390～454
34 S.R. Athreya, R.F. Bass, M. Gordina, E.A. Perkins. Infinite Dimensional Stochas-tic Differential Equations of Ornstein-Uhlenbeck Type. Stochastic Processes andtheir Applications. 2006, 116(3):381～406
35让光林.一类非Lipschitz条件下的量子随机微分方程解的存在唯一性.湖北大学学报(自然科学版). 2004, 26(2):9～101
36刘经洪,朱起定. Poisson积分方程的解的存在唯一性.湖南师范大学自然科学学报. 2004, 27(1):9～11,22
37胡良剑,赵伟国,冯玉瑚.伊藤型模糊随机微分方程.工程数学学报. 2006,23(1):52～62
38吴玥,孙晓君.一类倒向随机微分方程解的存在唯一性和稳定性.纺织高校基础科学学报. 2003, 16(2):134～147
39蒋达清,张宝学,王德辉,史宁中.具有随机扰动的Logistic方程正解的存在唯一性,全局吸引性及其参数的极大似然估计.中国科学(A辑). 2007,37(6):742～750
40 J. Yin, Y. Wang. Hilbert Space-valued Forward-backward Stochastic Differen-tial Equations with Poisson Jumps and Applications. Journal of MathematicalAnalysis and Applications. 2007, 328(1):438～451
41 D. Guo, S. Ji, H. Zhao. On the Solvability of Infinite Horizon Forward-backwardStochastic Differential Equations with absorption Coefficients. Statistics & Prob-ability Letters. 2006, 76(18):1954～1960
42 F. Delarue, G. Guatteri. Weak Existence and Uniqueness for Forward-backwardSDEs. Stochastic Processes and their Applications. 2006, 116(12):1712～1742
43 N.I. Mahmudov, M.A. McKibben. On Backward Stochastic Evolution Equa-tions in Hilbert Spaces and Optimal Control. Nonlinear Analysis, 2006,67(4):1260～1274
44王赢,王向荣.一类非Lipschitz条件的BSDE适应解的存在唯一性.应用概率统计. 2003, 19(3):245～251
45李森林,温立志.泛函微分方程.湖南科技出版社, 1986:56～105
46迪申加卜.具有无限时滞中立型泛函微分方程解的有界性.工程数学学报.2004, 21(04):631～634,638
47徐勇.具有无限时滞的随机泛函微分方程的解的唯一存在性.应用数学.2007, 20(4):830～836
2 T.C. Gard. Introduction to stochastic differential Equations. Marcel Dekker, NewYork, 1988:52～89
3 A. Friedman. Stochastic Differential Equations and Their Applications. Aca-demic Press, San Diego, 1976:39～120
4 Bernt, Ksendal. Stochastic Differential Equations. 6th Edition. Springer-VerlagBerlin, 2005:63～96
5东北师范大学微分方程教研室.常微分方程(第二版).高等教育出版社,2005:109～114
6尤秉礼.常微分方程补充教程.人民教育出版社, 1981:23～98
7 J. Hale, S.M.V. Lunel. Introduction to Functions Differential Equatons. NewYork:Springer, 1993:13～69
8 J. Littner. The Valuation of Risk Assets and the Selection of Risk Investments inStock Portfolios and Capital Budgets. Review of Economics and Statistic. 1965,47:13～37
9 S. Ross. The Arbitrage Theory of Capital Asset Pricing. Journal of EconomicTheory. 1976, 13(3):341～360
10 RC. Merton. The Theory of Rational Option Pricing. Bell Journal of Economics& Manag. 1973, 49:143～183
11 F. Wei, K. Wang. The Existence and Uniqueness of the Solution for StochasticFunctional Differential Equations with Infinite Delay. Journal of MathematicalAnalysis and Applications. 2007, 331(1):516～531
12 I. Gikhman, A. Skorokhood. Stochastc Differential Equations. Springer-Verlag,New York, 1974:35～87
13 L. Arnold. Stochastic Differential Equations Theory and Applications. Wily, NewYork, 1974:39～86
14 L. Arnold. Random Dynamical Systems. Spring-Verlag, 1998:57～110
15 EI Karoui N, S. Peng, M C. Quence. Backward stochastic differential equationin finance. Math Finance. 1997, 7(1):1～13
16 X. Mao. Stotistics Functional Differential Equations. Springer, 1976:36～81
17 T.C. Gard. Pathwise Uniqueness Solutions of Systems Integrals and Equations.Stochastic Process and Application. 1978, 6:253～260
18 X. Mao. Existence and Uniqueness of the Solutions of Delay Differential Equa-tions. Stochastic Analysis and Applications. 1989, 7(1):59～74
19 T.C. Gard. A General Uniqueness for Theory for Stochastic Differential Equa-tions. SIAM Journal on Control and Optimization. 1976, 14:445～457
20周少波,薛明皋.无限时滞中立型随机泛函微分方程解的存在唯一性.应用数学. 2008, 28(1):75～83
21 G. Cao. Existence and Pathwise Uniqueness of Solutions to Stochastic Dif-ferential Equations in Banach Space. Mathematica Applicata(China). 2006,19(1):75～79
22冉启康.一类非Lipschitz条件的BSDE解的存在唯一性.工程数学学报.2006, 23(2):286～292
23曹桂兰,何凯.由无穷个Brown单驱动的随机微分方程解的存在唯一性.数学物理学报. 2006, 26(6):813～823
24孙晓君.非Lipschitz条件下多维随机微分方程强解的存在唯一性.工程数学学报. 1998, 01:121～124
25聂赞坎.平面中随机微分方程解的存在唯一性.高校应用数学学报A辑(中文版). 1987, 2(2):222～227
26汪家义,许明浩.随机微分-积分方程解的存在唯一性.数学杂志. 1999,03:333～338
27史树中.凸分析.上海科学技术出版社, 1990:93～123
28 J. Martin Lindsay, Adam G. Skalski. On Quantum Stochastic Differen-tial Equations. Journal of Mathematical Analysis and Applications. 2007,330(2):1093～1114
29 F. Wang. Ito? type Measure-valued Stochastic Differential Equations. Journal ofMathematical Analysis and Applications. 2007, 329(2):1102～1117
30 H. Schurz. Existence and Uniqueness of Solutions of Semilinear Stochas-tic Infinite-Dimensional Differential Systems with H-regular Noise. Journal ofMathematical Analysis and Applications. 2006, 332(1):334～345
31 M. Rei, M. Riedle, O. van Gaans. Delay Differential Equations Driven By LévyProcesses: Stationarity and Feller Properties. Stochastic Processes and their Ap-plications. 2006, 11610:1409～1432
32 Y. Shen, L. Qi, X. Mao. The Improved LaSalle-type Theorems for StochasticFunctional Differential Equations. Journal of Mathematical Analysis and Appli-cations. 2006, 318(1):134～154
33 D. Nualart, P.A. Vuillermot. Variational Solutions for Partial Differential Equa-tions Driven By a Fractional Noise. Journal of Functional Analysis. 2006,232(2):390～454
34 S.R. Athreya, R.F. Bass, M. Gordina, E.A. Perkins. Infinite Dimensional Stochas-tic Differential Equations of Ornstein-Uhlenbeck Type. Stochastic Processes andtheir Applications. 2006, 116(3):381～406
35让光林.一类非Lipschitz条件下的量子随机微分方程解的存在唯一性.湖北大学学报(自然科学版). 2004, 26(2):9～101
36刘经洪,朱起定. Poisson积分方程的解的存在唯一性.湖南师范大学自然科学学报. 2004, 27(1):9～11,22
37胡良剑,赵伟国,冯玉瑚.伊藤型模糊随机微分方程.工程数学学报. 2006,23(1):52～62
38吴玥,孙晓君.一类倒向随机微分方程解的存在唯一性和稳定性.纺织高校基础科学学报. 2003, 16(2):134～147
39蒋达清,张宝学,王德辉,史宁中.具有随机扰动的Logistic方程正解的存在唯一性,全局吸引性及其参数的极大似然估计.中国科学(A辑). 2007,37(6):742～750
40 J. Yin, Y. Wang. Hilbert Space-valued Forward-backward Stochastic Differen-tial Equations with Poisson Jumps and Applications. Journal of MathematicalAnalysis and Applications. 2007, 328(1):438～451
41 D. Guo, S. Ji, H. Zhao. On the Solvability of Infinite Horizon Forward-backwardStochastic Differential Equations with absorption Coefficients. Statistics & Prob-ability Letters. 2006, 76(18):1954～1960
42 F. Delarue, G. Guatteri. Weak Existence and Uniqueness for Forward-backwardSDEs. Stochastic Processes and their Applications. 2006, 116(12):1712～1742
43 N.I. Mahmudov, M.A. McKibben. On Backward Stochastic Evolution Equa-tions in Hilbert Spaces and Optimal Control. Nonlinear Analysis, 2006,67(4):1260～1274
44王赢,王向荣.一类非Lipschitz条件的BSDE适应解的存在唯一性.应用概率统计. 2003, 19(3):245～251
45李森林,温立志.泛函微分方程.湖南科技出版社, 1986:56～105
46迪申加卜.具有无限时滞中立型泛函微分方程解的有界性.工程数学学报.2004, 21(04):631～634,638
47徐勇.具有无限时滞的随机泛函微分方程的解的唯一存在性.应用数学.2007, 20(4):830～836