IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i16p2436-d1450791.html
   My bibliography  Save this article

The Implicit Euler Scheme for FSDEs with Stochastic Forcing: Existence and Uniqueness of the Solution

Author

Listed:
  • Kęstutis Kubilius

    (Faculty of Mathematics and Informatics, Vilnius University, Akademijos g. 4, LT-08412 Vilnius, Lithuania)

Abstract

In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler’s approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall.

Suggested Citation

  • Kęstutis Kubilius, 2024. "The Implicit Euler Scheme for FSDEs with Stochastic Forcing: Existence and Uniqueness of the Solution," Mathematics, MDPI, vol. 12(16), pages 1-18, August.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:16:p:2436-:d:1450791
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/16/2436/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/16/2436/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. José Luís Silva & Mohamed Erraoui & El Hassan Essaky, 2018. "Mixed Stochastic Differential Equations: Existence and Uniqueness Result," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1119-1141, June.
    2. Li, Zhi & Zhan, Wentao & Xu, Liping, 2019. "Stochastic differential equations with time-dependent coefficients driven by fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 530(C).
    3. Kubilius, K., 2002. "The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 289-315, April.
    4. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kęstutis Kubilius, 2024. "Approximation of the Fractional SDEs with Stochastic Forcing," Mathematics, MDPI, vol. 12(24), pages 1-22, December.

    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. 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.
    2. David Baños & Salvador Ortiz-Latorre & Andrey Pilipenko & Frank Proske, 2022. "Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise," Journal of Theoretical Probability, Springer, vol. 35(2), pages 714-771, June.
    3. Mishura, Yu. & Nualart, D., 2004. "Weak solutions for stochastic differential equations with additive fractional noise," Statistics & Probability Letters, Elsevier, vol. 70(4), pages 253-261, December.
    4. Kubilius, K. & Skorniakov, V., 2016. "On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 159-167.
    5. Falkowski, Adrian & Słomiński, Leszek, 2022. "SDEs with two reflecting barriers driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 164-186.
    6. Gassiat, Paul & Mądry, Łukasz, 2023. "Perturbations of singular fractional SDEs," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 137-172.
    7. Boufoussi, Brahim & Hajji, Salah, 2012. "Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1549-1558.
    8. Marc Mukendi Mpanda, 2022. "Malliavin differentiability of fractional Heston-type model and applications to option pricing," Papers 2207.10709, arXiv.org, revised Aug 2022.
    9. Fan, Xiliang & Yu, Ting & Yuan, Chenggui, 2023. "Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 383-415.
    10. Xu, Liping & Luo, Jiaowan, 2018. "Stochastic differential equations driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 102-108.
    11. León, Jorge A. & Villa, José, 2011. "An Osgood criterion for integral equations with applications to stochastic differential equations with an additive noise," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 470-477, April.
    12. José Luís Silva & Mohamed Erraoui & El Hassan Essaky, 2018. "Mixed Stochastic Differential Equations: Existence and Uniqueness Result," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1119-1141, June.
    13. Slominski, Leszek & Ziemkiewicz, Bartosz, 2005. "Inequalities for the norms of integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 73(1), pages 79-90, June.
    14. Shevchenko, Georgiy & Shalaiko, Taras, 2013. "Malliavin regularity of solutions to mixed stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2638-2646.
    15. Azmoodeh Ehsan & Mishura Yuliya & Valkeila Esko, 2009. "On hedging European options in geometric fractional Brownian motion market model," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 129-144, December.
    16. Solesne Bourguin & Thanh Dang & Konstantinos Spiliopoulos, 2023. "Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-57, March.
    17. Sun, Xiaoxia & Guo, Feng, 2015. "On integration by parts formula and characterization of fractional Ornstein–Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 170-177.
    18. Li, Zhi & Yan, Litan, 2018. "Harnack inequalities for SDEs driven by subordinator fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 45-53.
    19. Vaidya, Tushar & Chotibut, Thiparat & Piliouras, Georgios, 2021. "Broken detailed balance and non-equilibrium dynamics in noisy social learning models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 570(C).
    20. Coffie, Emmanuel & Duedahl, Sindre & Proske, Frank, 2023. "Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 156-195.

    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:gam:jmathe:v:12:y:2024:i:16:p:2436-:d:1450791. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.