Convergence of θ-Milstein Method for Stochastic Differential Equations Driven by G-Brownian Motion

Although numerical methods for classical stochastic differential equations (SDEs) driven by Brownian motion are well-established, research on numerical schemes for SDEs driven by G-Brownian motion (referred to as G-SDEs) remains limited. Most existing studies are confined to Euler-Maruyama-type methods, which achieve only a strong convergence order of one-half. To bridge this gap, this paper aims to develop higher-order numerical methods for G-SDEs. By combining the classical Milstein method with the G-Itô formula, we propose a novel θ-Milstein scheme for G-SDEs. Using tools from G-expectation theory and Taylor expansions, we prove that the proposed scheme achieves a strong convergence order of one under the Lr-norm, assuming Lipschitz conditions. Numerical experiments demonstrate that the θ-Milstein method yields smaller errors and attains a higher convergence order compared to the Euler-Maruyama method, confirming its effectiveness and potential for advancing numerical solutions of G-SDEs.