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The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions

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  • Pengfei Zhao

    (Department of Mathematics, Harbin University, Harbin 150086, China)

  • Haiyan Yuan

    (Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China)

Abstract

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results.

Suggested Citation

  • Pengfei Zhao & Haiyan Yuan, 2025. "The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions," Mathematics, MDPI, vol. 13(12), pages 1-24, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:1993-:d:1680659
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    References listed on IDEAS

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    1. Wang, Feng-Yu, 2018. "Distribution dependent SDEs for Landau type equations," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 595-621.
    2. Gao, Fuqing, 2009. "Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3356-3382, October.
    3. Adams, Daniel & dos Reis, Gonçalo & Ravaille, Romain & Salkeld, William & Tugaut, Julian, 2022. "Large Deviations and Exit-times for reflected McKean–Vlasov equations with self-stabilising terms and superlinear drifts," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 264-310.
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