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Numerical Algorithms

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24-05-2019 | Original Paper

Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations

Authors: Yanan Jiang, Lihui Weng, Wei Liu

Published in: Numerical Algorithms | Issue 4/2020

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Abstract

The existence and uniqueness of the stationary distribution of the numerical solution generated by the stochastic theta method are studied. When the parameter theta takes different values, the requirements on the drift and diffusion coefficients are different. The convergence of the numerical stationary distribution to the true counterpart is investigated. Several numerical experiments are presented to demonstrate the theoretical results.

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Abdulle, A., Almuslimani, I., Vilmart, G.: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 6(2), 937–964 (2018)MathSciNetMATHCrossRef

Bao, J., Shao, J., Yuan, C.: Approximation of invariant measures for regime-switching diffusions. Potential Anal. 44(4), 707–727 (2016)MathSciNetMATHCrossRef

Berkolaiko, G., Buckwar, E., Kelly, C., Rodkina, A.: Almost sure asymptotic stability analysis of the 𝜃-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations. LMS J. Comput. Math. 15, 71–83 (2012)MathSciNetMATHCrossRef

Chen, L., Wu, F.: Almost sure exponential stability of the 𝜃-method for stochastic differential equations. Statist. Probab. Lett. 82(9), 1669–1676 (2012)MathSciNetMATHCrossRef

Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)MathSciNetMATHCrossRef

Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000). (electronic)MathSciNetMATHCrossRef

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. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130), 1563–1576 (2011)MathSciNetMATHCrossRef

Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, Volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1981)

Li, X., Ma, Q., Yang, H., Yuan, C.: The numerical invariant measure of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal. 56(3), 1435–1455 (2018)MathSciNetMATHCrossRef

10.

Li, X., Mao, X., Yin, G.: Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: Truncation methods, convergence in pth moment and stability. IMA J. Numer. Anal. 39(2), 847–892 (2019)MathSciNetCrossRef

11.

Liu, W., Mao, X.: Numerical stationary distribution and its convergence for nonlinear stochastic differential equations. J. Comput. Appl. Math. 276, 16–29 (2015)MathSciNetMATHCrossRef

12.

Mao, X.: Stability of Stochastic Differential Equations with Respect to Semimartingales, Volume 251 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1991)

13.

Mao, X., Yuan, C., Yin, G.: Numerical method for stationary distribution of stochastic differential equations with Markovian switching. J. Comput. Appl Math. 174(1), 1–27 (2005)MathSciNetMATHCrossRef

14.

Massey, F.J.: The Kolmogorov-Smirnov test for goodness of fit. J. Amer. Statist. Assoc. 46(253), 68–78 (1951)MATHCrossRef

15.

Qiu, Q., Liu, W., Hu, L.: Asymptotic moment boundedness of the stochastic theta method and its application for stochastic differential equations. Adv Difference Equ. 310(14), 2014 (2014)MathSciNetMATH

16.

Rodkina, A., Schurz, H.: Almost sure asymptotic stability of drift-implicit 𝜃-methods for bilinear ordinary stochastic differential equations in R¹. J. Comput. Appl Math. 180(1), 13–31 (2005)MathSciNetMATHCrossRef

17.

Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1973). Mathematics in Science and Engineering, vol. 103MATH

18.

Talay, D.: Second-order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics 29(1), 13–36 (1990)MATH

19.

Wang, W., Wen, L., Li, S.: Nonlinear stability of 𝜃-methods for neutral differential equations in Banach space. Appl. Math Comput. 198(2), 742–753 (2008)MathSciNetMATH

20.

Weng, L., Liu, W.: Invariant measures of the Milstein method for stochastic differential equations with commutative noise. Appl. Math Comput. 358, 169–176 (2019)MathSciNetMATHCrossRef

21.

Yang, H., Li, X.: Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons. J. Diff. Equ. 265(7), 2921–2967 (2018)MathSciNetMATHCrossRef

22.

Yuan, C., Mao, X.: Stability in distribution of numerical solutions for stochastic differential equations. Stochastic Anal. Appl. 22(5), 1133–1150 (2004)MathSciNetMATHCrossRef

23.

Yuan, C., Mao, X.: Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence. J. Difference Equ. Appl. 11(1), 29–48 (2005)MathSciNetMATHCrossRef

24.

Zong, X., Wu, F.: Choice of 𝜃 and mean-square exponential stability in the stochastic theta method of stochastic differential equations. J. Comput. Appl Math. 255, 837–847 (2014)MathSciNetMATHCrossRef

25.

Zong, X., Wu, F., Huang, C.: Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients. J. Comput. Appl Math. 278, 258–277 (2015)MathSciNetMATHCrossRef

Title: Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations
Authors: Yanan Jiang
Lihui Weng
Wei Liu
Publication date: 24-05-2019
Publisher: Springer US
Published in: Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-019-00735-5

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