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A Strong Averaging Principle Rate for Two-Time-Scale Coupled Forward–Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

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Abstract

This paper concerns the strong convergence rate of an averaging principle for two-time-scale coupled forward–backward stochastic differential equations (CFBSDEs, for short) driven by fractional Brownian motion (fBm, for short). The fast component is a forward stochastic differential equation (FSDE, for short) driven by Brownian motion, while the slow component is a backward stochastic differential equation (BSDE, for short) driven by fBm with the Hurst index greater than 1/2. Combining Malliavin calculus theory with stochastic integral and Khasminskii’s time discretization method, the rate of strong convergence for the slow component towards the solution of the averaging equation in the mean square sense is derived. The strong convergence rate of an averaging principle for fast–slow CFBSDEs driven by fBm is new.

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Acknowledgements

The first author acknowledges Professor Jinqiao Duan careful reading of manuscript, correcting errors, detailed comments and valuable suggestions, which improve the quality of this paper. The first author also thanks the support provided by NSFs of China No.11971154.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Jie Xu & Qiqi Lian

  2. Department of Mathematics, Computational Foundry, Swansea University, Swansea, SA1 8EN, UK

    Jiang-Lun Wu

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  1. Jie Xu
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  2. Qiqi Lian
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Correspondence to Jie Xu.

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Xu, J., Lian, Q. & Wu, JL. A Strong Averaging Principle Rate for Two-Time-Scale Coupled Forward–Backward Stochastic Differential Equations Driven by Fractional Brownian Motion. Appl Math Optim 88, 32 (2023). https://doi.org/10.1007/s00245-023-10008-2

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