Modified Runge–Kutta method with convergence analysis for nonlinear stochastic differential equations with Hölder continuous diffusion coefficient

Document Type : Research Article

Author

• A. Haghighi

Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran.

10.22067/ijnao.2022.78723.1181

Abstract

The main goal of this work is to develop and analyze an accurate trun-cated stochastic Runge–Kutta (TSRK2) method to obtain strong numeri-cal solutions of nonlinear one-dimensional stochastic differential equations (SDEs) with continuous Hölder diffusion coefficients. We will establish the strong L1-convergence theory to the TSRK2 method under the local Lipschitz condition plus the one-sided Lipschitz condition for the drift co-efficient and the continuous Hölder condition for the diffusion coefficient at a time T and over a finite time interval [0, T ], respectively. We show that the new method can achieve the optimal convergence order at a finite time T compared to the classical Euler–Maruyama method. Finally, nu-merical examples are given to support the theoretical results and illustrate the validity of the method.

Keywords

• Stochastic differential equation
• Strong convergence
• Truncated methods
• Hölder continuous coefficient

20.1001.1.24236977.2023.13.2.8.1

Main Subjects

• Numerical Analysis

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