Split-step balanced θ-method for SDEs with non-globally Lipschitz continuous coefficients

• Published in: Applied mathematics and computation 2022-01, Vol.413, p.126437, Article 126437
• Main Authors: Liu, Yufen, Cao, Wanrong, Li, Yuelin
• Format: Article
• Language: English
• Subjects: Exponential stability; Mean-square contraction; Nonlinear problems; Strong convergence; The balanced method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2021.126437

•It is proved that under some non-global Lipschitz conditions the SSBT scheme is convergent with order 0.5 in strong sense.•The presented SSBT scheme can preserve the exponential stability of the exact solution.•The technique we presented in Lemma 4.2 and Theorem 4.4 can be generalized for proving stability of other schemes. In this paper, a split-step balanced θ-method (SSBT) has been presented for solving stochastic differential equations (SDEs) under non-global Lipschitz conditions, where θ∈[0,1] is a parameter of the scheme. The moment boundedness and strong convergence of the numerical solution have been studied, and the convergence rate is 0.5. Moreover, under some conditions it is proved that the SSBT scheme can preserve the exponential mean-square stability of the exact solution when θ∈(1/2,1] for every step size h>0. Numerical examples verify the theoretical findings.