Modified stochastic theta methods by ODEs solvers for stochastic differential equations

• Published in: Communications in nonlinear science & numerical simulation Vol. 68; pp. 336 - 346
• Main Authors: Nouri, Kazem, Ranjbar, Hassan, Torkzadeh, Leila
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2019; Elsevier Science Ltd
• Subjects: Approximation; Corrections; Differential equations; Error correction; Mathematical analysis; Mean square errors; Mean-square convergence and stability; Numerical methods; ODEs solver; Ordinary differential equations; Solvers; Stability analysis; Stochastic differential equations; Stochastic models; Stochastic theta method; Taylor series; ODEs solver; Stochastic differential equations; 60H10; 41A25; Mean-square convergence and stability; 93E15; Stochastic theta method
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2018.08.013

•We proposed a class of improved stochastic theta methods, driven by the error corrected and exponential error corrected ODEs solvers, to solve a general class of linear and nonlinear stochastic differential equations.•Using the Itô–Taylor expansion under the Lipschitz conditions and linear growth bounds, we analyzed the mean–square convergence of the proposed scheme in the strong sense.•We investigated the numerical stability properties of a linear test equation with real parameters, based on the Descarte’s rule of signs, and sufficient conditions for the mean–square stability of solutions are provided.•Numerical examples are reported to confirm the theoretical results, and to illustrate the efficiency of the proposed methods for solving one and two dimensionals stochastic differential equations. In this paper, we present a family of stochastic theta methods modified by ODEs solvers for stochastic differential equations. This class of methods constructed by adding error correction and exponential error correction terms to the traditional stochastic theta methods. Using the Itô–Taylor expansion, analyzed mean-square convergence under the Lipschitz conditions and linear growth bounds. Also, we concern mean-square stability analysis of our proposed methods. Numerical examples are presented to demonstrate the efficiency of these methods for the pathwise approximation solution of some stochastic differential equations.