随机微分方程改进的分裂步单支θ方法的强收敛性
|
摘要
本文提出了一个改进的分裂步单支θ方法,在漂移项系数满足单边Lipschitz条件下,证明了当数值方法的参数θ满足1/2≤θ≤1时,该数值方法对于这类随机微分方程是强收敛的,并在现有文献的基础上将方法的收敛阶从1/2阶提高到1阶;当0≤θ≤1/2时,若漂移项系数进一步满足线性增长条件,该数值方法也是强收敛的,收敛阶为1阶.文末的数值试验验证了理论结果的正确性.
In this paper a class of methods, called improved split-step one-leg theta methods(ISSOLTM), are introduced and are shown to be convergent for SDEs with one-sided Lipschitz continuous drift coefficient if the method parameter satisfies 1/2 ≤θ≤1. At the same time, we improve the strong order from one half to one on the basis of the existing literature.For 0≤θ≤1/2, under the additional linear growth condition for the drift coefficient, the methods are also strongly convergent with the the order 1. Finally, the obtained results are supported by numerical experiments.
In this paper a class of methods, called improved split-step one-leg theta methods(ISSOLTM), are introduced and are shown to be convergent for SDEs with one-sided Lipschitz continuous drift coefficient if the method parameter satisfies 1/2 ≤θ≤1. At the same time, we improve the strong order from one half to one on the basis of the existing literature.For 0≤θ≤1/2, under the additional linear growth condition for the drift coefficient, the methods are also strongly convergent with the the order 1. Finally, the obtained results are supported by numerical experiments.
引文
[1] Wang X, Gan S. B-convergence of split-step one-leg theta methods for stochastic differential equations[J]. Journal of Applied Mathematics and Computing, 2012, 38(1):489-503.
[2] Friedman A. Stochastic Differential Equations and Applications[M]. Berlin Heidelberg:Springer,2010, 75-148.
[3] Oksendal B. Stochastic Differential Equations-An Introduction With Applications[M]. New York:Journal of the American Statistical Association, 2000, 11-32.
[4] Szpruch L. Numerical approximations of nonlinear stochastic systems[D]. University of Strathclyde, 2010.
[5]齐莎.非线性复合刚性发展方程正则Euler分裂方法[D].湘潭大学,2015.
[6]周梦.二维空间分数阶扩散方程的正则Euler分裂方法[D].湘潭大学,2016.
[7]任铭,程瑶,张永胜.一类偏微分方程的多分裂迭代并行解法[J].现代电子技术,2011,34(12):55-56.
[8]王文强,李寿佛,黄内山.非线性随机延迟微分方程半隐式Euler方法的收敛性[J].云南大学学报(自然科学版),2008, 30(1):11-15.
[9]谭英贤,甘四清,王小捷.随机延迟微分方程平衡方法的均方收敛性与稳定性[J].计算数学,2011,33(1):25-36.
[10]易玉连,王文强.Poisson跳的随机延迟微分方程Heun方法的均方收敛性[J].应用数学,2015, 28(4):938-948.
[11] Hu Y. Semi-implicit Euler-Maruyama scheme for stiff stochastic equations[M]. Stochastic Analysis and Related Topics V. Birkhauser Boston, 1996, 183-202.
[12] Maruyama G. Continuous Markov processes and stochastic equations[J]. Rendiconti del Circolo Matematico di Palermo, 1955, 4(1):48-90.
[13] 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[C].Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences.The Royal Society, 2011, 467(2130):1563-1576.
[14] Hutzenthaler M, Jentzen A, Kloeden P E. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients[J]. The Annals of Applied Probability,2012:1611-1641.
[15] Zong X, Wu F, Xu G. Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations[J]. arXiv preprint arXiv:1501.03695, 2015.
[16] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM review, 2001, 43(3):525-546.
[2] Friedman A. Stochastic Differential Equations and Applications[M]. Berlin Heidelberg:Springer,2010, 75-148.
[3] Oksendal B. Stochastic Differential Equations-An Introduction With Applications[M]. New York:Journal of the American Statistical Association, 2000, 11-32.
[4] Szpruch L. Numerical approximations of nonlinear stochastic systems[D]. University of Strathclyde, 2010.
[5]齐莎.非线性复合刚性发展方程正则Euler分裂方法[D].湘潭大学,2015.
[6]周梦.二维空间分数阶扩散方程的正则Euler分裂方法[D].湘潭大学,2016.
[7]任铭,程瑶,张永胜.一类偏微分方程的多分裂迭代并行解法[J].现代电子技术,2011,34(12):55-56.
[8]王文强,李寿佛,黄内山.非线性随机延迟微分方程半隐式Euler方法的收敛性[J].云南大学学报(自然科学版),2008, 30(1):11-15.
[9]谭英贤,甘四清,王小捷.随机延迟微分方程平衡方法的均方收敛性与稳定性[J].计算数学,2011,33(1):25-36.
[10]易玉连,王文强.Poisson跳的随机延迟微分方程Heun方法的均方收敛性[J].应用数学,2015, 28(4):938-948.
[11] Hu Y. Semi-implicit Euler-Maruyama scheme for stiff stochastic equations[M]. Stochastic Analysis and Related Topics V. Birkhauser Boston, 1996, 183-202.
[12] Maruyama G. Continuous Markov processes and stochastic equations[J]. Rendiconti del Circolo Matematico di Palermo, 1955, 4(1):48-90.
[13] 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[C].Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences.The Royal Society, 2011, 467(2130):1563-1576.
[14] Hutzenthaler M, Jentzen A, Kloeden P E. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients[J]. The Annals of Applied Probability,2012:1611-1641.
[15] Zong X, Wu F, Xu G. Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations[J]. arXiv preprint arXiv:1501.03695, 2015.
[16] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM review, 2001, 43(3):525-546.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |