Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance

Bou Diop

doi:doi:10.11648/j.ajam.20251306.17

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Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance

Bou Diop^*

Published in American Journal of Applied Mathematics (Volume 13, Issue 6)

Received: 16 September 2025 Accepted: 11 October 2025 Published: 19 December 2025

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Abstract

This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 6)
DOI	10.11648/j.ajam.20251306.17
Page(s)	452-461
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Fractional Stochastic Differential Equation, Fractional Brownian Motion, Jump Processes, Euler-Maruyama Method, Strong Convergence, Weak Convergence, Malliavin Calculus, Option Pricing

References

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[12]	Kloeden, P. E., & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer-Verlag.
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[19]	Alfonsi, A. (2010). High order discretization schemes for the CIR process: Application to affine term structure and Heston models. Mathematics of Computation, 79(269), 209-237.
[20]	Neuenkirch, A., & Nourdin, I. (2008). Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. Journal of Theoretical Probability, 21(3), 729-753.
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Diop, B. (2025). Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. American Journal of Applied Mathematics, 13(6), 452-461. https://doi.org/10.11648/j.ajam.20251306.17

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ACS Style

Diop, B. Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. Am. J. Appl. Math. 2025, 13(6), 452-461. doi: 10.11648/j.ajam.20251306.17

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AMA Style

Diop B. Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. Am J Appl Math. 2025;13(6):452-461. doi: 10.11648/j.ajam.20251306.17

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@article{10.11648/j.ajam.20251306.17,
  author = {Bou Diop},
  title = {Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {452-461},
  doi = {10.11648/j.ajam.20251306.17},
  url = {https://doi.org/10.11648/j.ajam.20251306.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.17},
  abstract = {This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.
},
 year = {2025}
}

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TY  - JOUR
T1  - Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance

AU  - Bou Diop
Y1  - 2025/12/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.17
DO  - 10.11648/j.ajam.20251306.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 452
EP  - 461
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.17
AB  - This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.

VL  - 13
IS  - 6
ER  -

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Bou Diop

Department of Mathematics, Iba Der Thiam University, Thiès, Senegal

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APA Style

Diop, B. (2025). Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. American Journal of Applied Mathematics, 13(6), 452-461. https://doi.org/10.11648/j.ajam.20251306.17

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ACS Style

Diop, B. Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. Am. J. Appl. Math. 2025, 13(6), 452-461. doi: 10.11648/j.ajam.20251306.17

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AMA Style

Diop B. Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance. Am J Appl Math. 2025;13(6):452-461. doi: 10.11648/j.ajam.20251306.17

Copy | Download

@article{10.11648/j.ajam.20251306.17,
  author = {Bou Diop},
  title = {Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {452-461},
  doi = {10.11648/j.ajam.20251306.17},
  url = {https://doi.org/10.11648/j.ajam.20251306.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.17},
  abstract = {This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance

AU  - Bou Diop
Y1  - 2025/12/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.17
DO  - 10.11648/j.ajam.20251306.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 452
EP  - 461
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.17
AB  - This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.

VL  - 13
IS  - 6
ER  -

Copy | Download