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Anticipated BSDEs Driven by Fractional Brownian Motion with a Time-Delayed Generator

Author

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  • Pei Zhang

    (Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia
    School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China)

  • Adriana Irawati Nur Ibrahim

    (Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia)

  • Nur Anisah Mohamed

    (Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia)

Abstract

This article describes a new form of an anticipated backward stochastic differential equation (BSDE) with a time-delayed generator driven by fractional Brownian motion, further known as fractional BSDE, with a Hurst parameter H ∈ ( 1 / 2 , 1 ) . This study expands upon the findings of the anticipated BSDE by considering the scenario when the driver is fractional Brownian motion rather instead of standard Brownian motion. Additionally, the generator incorporates not only the present and future but also the past. We will demonstrate the existence and uniqueness of the solutions to these equations by employing the fixed point theorem. Furthermore, an equivalent comparison theorem is derived.

Suggested Citation

  • Pei Zhang & Adriana Irawati Nur Ibrahim & Nur Anisah Mohamed, 2023. "Anticipated BSDEs Driven by Fractional Brownian Motion with a Time-Delayed Generator," Mathematics, MDPI, vol. 11(23), pages 1-13, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4845-:d:1292583
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    References listed on IDEAS

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    1. Tie Wang & Siyu Cui, 2022. "Anticipated Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients," Mathematics, MDPI, vol. 10(3), pages 1-18, January.
    2. Douissi, Soukaina & Wen, Jiaqiang & Shi, Yufeng, 2019. "Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 282-298.
    3. Wen, Jiaqiang & Shi, Yufeng, 2017. "Anticipative backward stochastic differential equations driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 118-127.
    4. Lucian Maticiuc & Tianyang Nie, 2015. "Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities," Journal of Theoretical Probability, Springer, vol. 28(1), pages 337-395, March.
    5. Ping He & Yong Ren & Defei Zhang, 2020. "A Study on a New Class of Backward Stochastic Differential Equation," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-9, May.
    6. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
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