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Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker–Planck equations for special stochastic process in foreign exchange markets

Author

Listed:
  • Habibi, Noora
  • Lashkarian, Elham
  • Dastranj, Elham
  • Hejazi, S. Reza

Abstract

In this paper the transition joint probability density function of the solution of the Ornstein–Uhlenbeck process is presented by a deterministic parabolic time-fractional PDE (FPDE), named time-fractional Fokker–Planck equation. The article generally is divided to two parts: theoretical and approximated analysis. In the theoretical sections Lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation. In the next part, by Chebyshev wavelets’s method the numerical solutions are driven. Then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs.

Suggested Citation

  • Habibi, Noora & Lashkarian, Elham & Dastranj, Elham & Hejazi, S. Reza, 2019. "Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker–Planck equations for special stochastic process in foreign exchange markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 750-766.
    • Handle: RePEc:eee:phsmap:v:513:y:2019:i:c:p:750-766
    DOI: 10.1016/j.physa.2018.08.155
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    References listed on IDEAS

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    1. Azadeh Naderifard & Elham Dastranj & S. Reza Hejazi, 2018. "Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 1-15, June.
    2. Smirnov, A.P. & Shmelev, A.B. & Sheinin, E.Ya., 2004. "Analysis of Fokker–Planck approach for foreign exchange market statistics study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 203-206.
    3. Lashkarian, Elham & Reza Hejazi, S., 2017. "Group analysis of the time fractional generalized diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 572-579.
    4. Zhi Mao & Aiguo Xiao & Zuguo Yu & Long Shi, 2014. "Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-11, July.
    5. Montagnon, Chris, 2015. "A closed solution to the Fokker–Planck equation applied to forecasting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 14-22.
    6. Tarasov, Vasily E. & Zaslavsky, George M., 2008. "Fokker–Planck equation with fractional coordinate derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6505-6512.
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    Cited by:

    1. Dastranj, Elham & Sahebi Fard, Hossein & Abdolbaghi, Abdolmajid & Reza Hejazi, S., 2020. "Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market)," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    2. Mohammadi, Shaban & Hejazi, S. Reza, 2023. "Using particle swarm optimization and genetic algorithms for optimal control of non-linear fractional-order chaotic system of cancer cells," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 538-560.

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