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Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries

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  • Azadeh Naderifard

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

  • Elham Dastranj

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

  • S. Reza Hejazi

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

Abstract

In this paper, the transition joint probability density function of the solution of geometric Brownian motion (GBM) equation is obtained via Lie group theory of differential equations (DEs). Lie symmetry analysis is applied to find new solutions for time-fractional Fokker–Planck–Kolmogorov equation of GBM. This analysis classifies the forms of the solutions for the equation by the similarity variables arising from the symmetry operators. Finally, an analytic method called invariant subspace method is applied in order to find another exact solution.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijfexx:v:05:y:2018:i:02:n:s2424786318500093
    DOI: 10.1142/S2424786318500093
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    References listed on IDEAS

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    1. Huang, Qing & Zhdanov, Renat, 2014. "Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 110-118.
    2. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
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    1. 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.
    2. 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).

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