IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v256y2015icp20-36.html
   My bibliography  Save this article

Fractional stochastic Volterra equation perturbed by fractional Brownian motion

Author

Listed:
  • Zhang, Yinghan
  • Yang, Xiaoyuan

Abstract

In this paper, we consider a class of fractional stochastic Volterra equation of convolution type driven by infinite dimensional fractional Brownian motion with Hurst index h∈(0,1). Base on the explicit formula for the scalar resolvent function and the properties of the Mittag–Leffler’s function, the existence and regularity results of the stochastic convolution process are established. Separate proofs are required for the cases of Hurst parameter above and below 12 and it will turn out that the regularity of the solution increases with Hurst parameter h.

Suggested Citation

  • Zhang, Yinghan & Yang, Xiaoyuan, 2015. "Fractional stochastic Volterra equation perturbed by fractional Brownian motion," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 20-36.
  • Handle: RePEc:eee:apmaco:v:256:y:2015:i:c:p:20-36
    DOI: 10.1016/j.amc.2015.01.046
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300315000600
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2015.01.046?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    2. Balan, Raluca M. & Tudor, Ciprian A., 2010. "The stochastic wave equation with fractional noise: A random field approach," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2468-2494, December.
    3. Grecksch, W. & Anh, V. V., 1999. "A parabolic stochastic differential equation with fractional Brownian motion input," Statistics & Probability Letters, Elsevier, vol. 41(4), pages 337-346, February.
    4. Quer-Sardanyons, Lluís & Tindel, Samy, 2007. "The 1-d stochastic wave equation driven by a fractional Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1448-1472, October.
    5. Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
    6. B. Pasik-Duncan & T. E. Duncan & B. Maslowski, 2006. "Linear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion," International Series in Operations Research & Management Science, in: Houmin Yan & George Yin & Qing Zhang (ed.), Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, chapter 0, pages 201-221, Springer.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Issoglio, E. & Riedle, M., 2014. "Cylindrical fractional Brownian motion in Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3507-3534.
    2. Lihong Guo, 2024. "Renormalization Group Method for a Stochastic Differential Equation with Multiplicative Fractional White Noise," Mathematics, MDPI, vol. 12(3), pages 1-20, January.
    3. Boufoussi, Brahim & Hajji, Salah, 2017. "Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 222-229.
    4. Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
    5. John-Fritz Thony & Jean Vaillant, 2022. "Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    6. B. L. S. Prakasa Rao, 2021. "Nonparametric Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion with Random Effects," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 554-568, August.
    7. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    8. Quer-Sardanyons, Lluís & Tindel, Samy, 2012. "Pathwise definition of second-order SDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 466-497.
    9. Falkowski, Adrian & Słomiński, Leszek, 2017. "SDEs with constraints driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3536-3557.
    10. Kęstutis Kubilius & Aidas Medžiūnas, 2020. "Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient," Mathematics, MDPI, vol. 9(1), pages 1-14, December.
    11. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.
    12. Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.
    13. Hong, Jialin & Huang, Chuying & Kamrani, Minoo & Wang, Xu, 2020. "Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2675-2692.
    14. Alexander Melnikov & Yuliya Mishura & Georgiy Shevchenko, 2015. "Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 17(1), pages 169-188, March.
    15. Balasubramaniam, P., 2022. "Solvability of Atangana-Baleanu-Riemann (ABR) fractional stochastic differential equations driven by Rosenblatt process via measure of noncompactness," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    16. Pavel Kříž & Leszek Szała, 2020. "The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
    17. Nualart, David & Pérez-Abreu, Victor, 2014. "On the eigenvalue process of a matrix fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4266-4282.
    18. Balan, Raluca M. & Tudor, Ciprian A., 2010. "The stochastic wave equation with fractional noise: A random field approach," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2468-2494, December.
    19. Elisa Alòs & Jorge A. León, 2021. "An Intuitive Introduction to Fractional and Rough Volatilities," Mathematics, MDPI, vol. 9(9), pages 1-22, April.
    20. Ahmadian, D. & Ballestra, L.V. & Shokrollahi, F., 2022. "A Monte-Carlo approach for pricing arithmetic Asian rainbow options under the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:256:y:2015:i:c:p:20-36. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.