The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions
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 c...
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| author | Pengfei Zhao Haiyan Yuan |
| author_facet | Pengfei Zhao Haiyan Yuan |
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| description | 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. |
| format | Article |
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| language | English |
| publishDate | 2025-06-01 |
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| spelling | doaj-art-9fe0b09266f84defa3510ff0c866c7a92025-08-20T02:20:58ZengMDPI AGMathematics2227-73902025-06-011312199310.3390/math13121993The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz ConditionsPengfei Zhao0Haiyan Yuan1Department of Mathematics, Harbin University, Harbin 150086, ChinaDepartment of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, ChinaIn 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.https://www.mdpi.com/2227-7390/13/12/1993G-MFSDEsG-Brownian motionmodified stochastic theta methodstrong convergenceasymptotic mean square stability |
| spellingShingle | Pengfei Zhao Haiyan Yuan The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions Mathematics G-MFSDEs G-Brownian motion modified stochastic theta method strong convergence asymptotic mean square stability |
| title | The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions |
| title_full | The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions |
| title_fullStr | The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions |
| title_full_unstemmed | The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions |
| title_short | The Modified Stochastic Theta Scheme for Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion Under Local One-Sided Lipschitz Conditions |
| title_sort | modified stochastic theta scheme for mean field stochastic differential equations driven by g brownian motion under local one sided lipschitz conditions |
| topic | G-MFSDEs G-Brownian motion modified stochastic theta method strong convergence asymptotic mean square stability |
| url | https://www.mdpi.com/2227-7390/13/12/1993 |
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