Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations
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- Math. Comp. 90 (2021), 2827-2872 Request permission
Abstract:
In this article we introduce a number of explicit schemes, which are amenable to Khasminski’s technique and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restrictions to those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov-type functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the convergence theorem of nonnegative semimartingales, positive results about the ability of the explicit numerical scheme proposed to reproduce the well-known LaSalle-type theorem of SDEs are proved here, from which we deduce the asymptotic stability of numerical solutions. Some examples are discussed to demonstrate the validity of the new numerical schemes and computer simulations are performed to support the theoretical results.References
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Additional Information
- Xiaoyue Li
- Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, People’s Republic of China
- Email: lixy209@nenu.edu.cn
- Xuerong Mao
- Affiliation: Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom
- MR Author ID: 199088
- ORCID: 0000-0002-6768-9864
- Email: x.mao@strath.ac.uk
- Hongfu Yang
- Affiliation: School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, People’s Republic of China
- Email: yanghf783@nenu.edu.cn
- Received by editor(s): December 20, 2018
- Received by editor(s) in revised form: June 3, 2020, and January 31, 2021
- Published electronically: June 22, 2021
- Additional Notes: The research of the first author was supported in part by the National Key R&D Program of China (No. 2020YFA0714102), the National Natural Science Foundation of China (11971096), the Natural Science Foundation of Jilin Province (YDZJ202101ZYTS154), the Education Department of Jilin Province (JJKH20211272KJ), the Fundamental Research Funds for the Central Universities. The research of the second author was supported in part by the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1)
The third author is the corresponding author - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2827-2872
- MSC (2020): Primary 65C30, 60H35, 65H10
- DOI: https://doi.org/10.1090/mcom/3661
- MathSciNet review: 4305371