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

The main goal of this work is to develop and analyze an accurate truncated stochastic Runge–Kutta (TSRK2) method to obtain strong numerical 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 coefficient 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, numerical examples are given to support the theoretical results and illustrate the validity of the method.