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高阶分裂步(θ1,θ2,θ3)方法的强收敛性

岳超   

  1. 郑州航空工业管理学院 经贸学院, 郑州 450015
  • 收稿日期:2017-04-29 出版日期:2019-06-15 发布日期:2019-05-18
  • 基金资助:

    国家自然科学基金(11371157,71603243),河南省高校重点科研项目(17A110013,17A520062),2016年河南省政府决策研究招标课题(2016B017,2016B013)和2017年度河南省科技攻关计划(高新技术领域)项目(172102210529)资助.

岳超. 高阶分裂步(θ1,θ2,θ3)方法的强收敛性[J]. 计算数学, 2019, 41(2): 126-155.

Chao Yue. STRONG CONVERGENCE OF HIGH-ORDER SPLIT-STEP (θ1, θ2, θ3) METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2019, 41(2): 126-155.

STRONG CONVERGENCE OF HIGH-ORDER SPLIT-STEP (θ1, θ2, θ3) METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Chao Yue   

  1. School of Economics and Trade, Zhengzhou University of Aeronautics, Zhengzhou 450015, China
  • Received:2017-04-29 Online:2019-06-15 Published:2019-05-18
本文首先提出一类高阶分裂步(θ1θ2θ3)方法求解由非交换噪声驱动的非自治随机微分方程.其次在漂移项系数满足多项式增长和单边Lipschitz条件下,证明了当1/2 ≤ θ2 ≤ 1时该方法是1阶强收敛的.此类方法包含很多经典的方法:如随机θ-Milstein方法,向后分裂步Milstein方法等.最后数值实验验证了所得结论.
In this paper, we first propose high-order split-step (θ1, θ2, θ3) methods for non-autonomous stochastic differential equations (SDEs) driven by non-commutative noise. Then, we prove that for 1/2 ≤ θ2 ≤ 1 the high-order split-step (θ1, θ2, θ3) methods are convergent with strong order of one for SDEs with the drift coefficient satisfying a superlinearly growing condition and a one-sided Lipschitz continuous condition. The high-order split-step (θ1, θ2, θ3) methods contain some classical methods such as stochastic θ-Milstein method, split-step back Milstein method and so on. Finally, the obtained results are verified by numerical experiments.
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[1] Sobczyk K. Stochastic differential equations:with applications to physics and engineering[M]. volume 40. Springer, 2001.

[2] Mel'nikov A V. Stochastic differential equations:singularity of coefficients, regression models, and stochastic approximation[J]. Russ. Math. Surv., 1996, 51(5):819-909.

[3] Yan L. Convergence of the Euler scheme for stochastic differential equations with irregular coefficients[J]. ETD Collection for Purdue University, 2000, page AAI3018308.

[4] Situ R. Theory of stochastic differential equations with jumps and applications[M]. volume 40. Springer-Verlag:Berlin, 2010.

[5] Kloeden P E, Platen E. Numerical solution of stochastic differential equations[M]. volume 23. Springer-Verlag:Berlin, 1992.

[6] Huang C. Exponential mean square stability of numerical methods for systems of stochastic differential equations[J]. J. Comput. Appl. Math., 2012, 236(16):4016-4026.

[7] Guo Q, Li H, Zhu Y. The improved split-step θ methods for stochastic differential equation[J]. Math. Meth. Appl. Sci., 2014, 37(15):2245-2256.

[8] Ding X, Ma Q, Zhang L. Convergence and stability of the split-step θ-method for stochastic differential equations[J]. Comput. Math. Appl., 2010, 60(5):1310-1321.

[9] Hu Y. Semi-implicit Euler-Maruyama scheme for stiff stochastic equations[G]. In Stochastic Analysis and Related Topics V, pages 183-202. Birkhäuser Boston, 1996.

[10] Higham D J, Mao X, Stuart A M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations[J]. SIAM J. Numer. Anal., 2002, 40(3):1041-1063.

[11] Maruyama G. Continuous Markov processes and stochastic equations[J]. Rend. Circolo. Math. Palermo, 1955, 4(1):48-90.

[12] Hutzenthaler M, Jentzen A, Kloeden P E. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Proc. R. Soc. A-Math. Phys. Eng. Sci., 2011, 467(2130):1563-1576.

[13] Hutzenthaler M, Jentzen A, Kloeden P E. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients[J]. Ann. Appl. Probab., 2012, 22(4):1611-1641.

[14] Wang X, Gan S. B-convergence of split-step one-leg theta methods for stochastic differential equations[J]. J.Appl. Math. Comput., 2012, 38(1-2):489-503.

[15] Yue C, Huang C, Jiang F. Strong convergence of split-step theta methods for non-autonomous stochastic differential equations[J]. Int. J. Comput. Math., 2014, 91(10):2260-2275.

[16] Mao X, Szpruch L. Strong convergence rates for backward Euler-Maruyama method for nonlinear dissipative-type stochastic differential equations with super-linear diffusion coefficients[J]. Stochastics, 2013, 85(1):144-171.

[17] Tretyakov M, Zhang Z. A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications[J]. SIAM J. Numer. Anal., 2013, 51(6):3135-3162.

[18] Wang X, Gan S. The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. J. Differ. Equ. Appl., 2013, 19(3):466-490.

[19] Jiang F, Zong X, Yue C, Huang C. Double-implicit and split two-step Milstein schemes for stochastic differential equations[J]. Int. J. Comput. Math., 2016, 93(12):1987-2011.

[20] Zong X, Wu F, Xu G. Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations[J]. Mathematics, 2015, 17(2).

[21] Yue C. High-order split-step theta methods for non-autonomous stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Math. Meth. Appl. Sci., 2016, 39(9):2380-2240.

[22] Higham D J, Mao X, Szpruch L. Convergence, non-negativity and stability of a new milstein scheme with applications to finance[J]. Discrete Contin. Dyn. Syst.-Ser. B, 2013, 18(8):2083-2100.

[23] Burrage K, Burrage P. High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula[J]. PHYSICA D, 1999, 133(1):34-48.

[24] Magnus W. On the exponential solution of differential equations for a linear operator[J]. Communications on pure and applied mathematics, 1954, 7(4):649-673.

[25] Buckwar E, Sickenberger T. A structural analysis of asymptotic mean-square stability for multidimensional linear stochastic differential systems[J]. Appl. Numer. Math., 2012, 62(7):842-859.

[26] Reshniak V, Khaliq A, Voss D, Zhang G. Split-step Milstein methods for multi-channel stiff stochastic differential systems[J]. Appl. Numer. Math., 2015, 89:1-23.

[27] Mao X. stochastic diffrential equations and applications[M]. Horwood Publishing:Chichester,, 1997.

[28] Ambrosetti A, Prodi G. A primer of nonlinear analysis[M]. volume 34. Cambridge University Press, 1995.

[29] 王鹏. 随机常微分方程数值分析中的若干方法[D]. 吉林大学, 2008.

[30] 王鹏, 韩月才. 求解刚性随机系统的分步向后Milstein方法[J]. 吉林大学学报:理学版, 2009, 47(6):1150-1154.

[31] Kloeden P E, Platen E, Wright I. The approximation of multiple stochastic integrals[J]. Stoch. Anal. Appl., 1992, 10(4):431-441.

[32] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Rev., 2001, 43(3):525-546.
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