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 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.