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

Lipschitz condition
θ-Milstein scheme
G-Brownian motion
Strong convergence
Stochastic differential equations

How to Cite

Deng, S., Shen, B., & Fei, W. (2025). Convergence of θ-Milstein Method for Stochastic Differential Equations Driven by G-Brownian Motion. Journal of Advances in Applied & Computational Mathematics, 12, 82–106. https://doi.org/10.15377/2409-5761.2025.12.7

Abstract

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.

https://doi.org/10.15377/2409-5761.2025.12.7
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Copyright (c) 2025 Shounian Deng, Bo Shen, Weiyin Fei

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