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Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations

Part of:Probabilistic methods, simulation and stochastic differential equationsStochastic analysis

Published online by Cambridge University Press:  12 September 2017

Yabing Sun,
Jie Yang and
Weidong Zhao
Yabing Sun*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China
Jie Yang*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China
Weidong Zhao*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China
*
*Corresponding author. Email addresses:sunybly@163.com (Y. B. Sun), yangjie218@mail.sdu.edu.cn (J. Yang), wdzhao@sdu.edu.cn (W. D. Zhao)
*Corresponding author. Email addresses:sunybly@163.com (Y. B. Sun), yangjie218@mail.sdu.edu.cn (J. Yang), wdzhao@sdu.edu.cn (W. D. Zhao)
*Corresponding author. Email addresses:sunybly@163.com (Y. B. Sun), yangjie218@mail.sdu.edu.cn (J. Yang), wdzhao@sdu.edu.cn (W. D. Zhao)

Abstract

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This paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

Keywords

Itô-Taylor schememean-field stochastic differential equationmean-field Itô-Taylor formulaerror estimate

MSC classification

Secondary:60H35: Computational methods for stochastic equations65C30: Stochastic differential and integral equations60H10: Stochastic ordinary differential equations

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017, pp. 798 - 828
Copyright
Copyright © Global-Science Press 2017 

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