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Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications

Author

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  • Reem Abdullah Aljethi

    (Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh 11564, Saudi Arabia
    Department of Mathematics and Statistics, University Putra Malaysia, Serdang 43400, Selangor, Malaysia)

  • Adem Kılıçman

    (Department of Mathematics and Statistics, University Putra Malaysia, Serdang 43400, Selangor, Malaysia)

Abstract

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model , as well as develop an R code that can be used to draw figures from real data.

Suggested Citation

  • Reem Abdullah Aljethi & Adem Kılıçman, 2023. "Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications," Mathematics, MDPI, vol. 11(5), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1102-:d:1077277
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    References listed on IDEAS

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    1. Palatella, Luigi & Perelló, Josep & Montero, Miquel & Masoliver, Jaume, 2005. "Diffusion Entropy technique applied to the study of the market activity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(1), pages 131-137.
    2. Prehl, J. & Essex, C. & Hoffmann, K.H., 2010. "The superdiffusion entropy production paradox in the space-fractional case for extended entropies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(2), pages 215-224.
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