American Institute of Mathematical Sciences

Advanced
September  2010, 14(2): 389-407. doi: 10.3934/dcdsb.2010.14.389

Two-sided error estimates for the stochastic theta method

Wolf-Jürgen Beyn 1, and  Raphael Kruse 1,

1. 

Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany

Received  April 2009 Revised  August 2009 Published  June 2010

Two-sided error estimates are derived for the strong error of convergence of the stochastic theta method. The main result is based on two ingredients. The first one shows how the theory of convergence can be embedded into standard concepts of consistency, stability and convergence by an appropriate choice of norms and function spaces. The second one is a suitable stochastic generalization of Spijker's norm (1968) that is known to lead to two-sided error estimates for deterministic one-step methods. We show that the stochastic theta method is bistable with respect to this norm and that well-known results on the optimal $\mathcal{O}(\sqrt{h})$ order of convergence follow from this property in a natural way.
Keywords: stochastic Spijker norm., two-sided error estimate, stochastic theta method, SODE, consistency, bistability.
Mathematics Subject Classification: Primary 65C20, 65C30, 65L20 ; Secondary 65L70, 65J1.
Citation: Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389
[1]

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
[2]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[3]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339
[4]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
[5]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203
[6]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585
[7]

Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1081-1095. doi: 10.3934/mcrf.2018046
[8]

JaEun Ku. Maximum norm error estimates for Div least-squares method for Darcy flows. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1305-1318. doi: 10.3934/dcds.2010.26.1305
[9]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[10]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
[11]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic & Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003
[12]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093
[13]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[14]

Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373
[15]

Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016
[16]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[17]

Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35
[18]

Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092
[19]

Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529
[20]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences