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Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

Author

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  • Jonathan Blackledge

    (Stokes Professor, Science Foundation Ireland, Three Park Place, Dublin 2, Ireland
    Honorary Professor, School of Electrical and Electronic Engineering, Technological University, Kevin Street, Dublin 8, Ireland
    Professor Extraordinaire, Department of Computer Science, University of Western Cape, Bellville 7535, Cape Town, South Africa
    Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Westville Campus, University Road, Westville 3630, Durban, South Africa)

  • Derek Kearney

    (Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
    Lecturer, School of Electrical and Electronic Engineering, Technological University Dublin, Kevin Street, Dublin 8, Ireland)

  • Marc Lamphiere

    (Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
    Research Associate, School of Electrical and Electronic Engineering, Technological University Dublin, Kevin Street, Dublin 8, Ireland
    Country Director, The Natural Power Consultants (Ireland) Limited, Beacon Court Sandyford, Dublin 18, Ireland)

  • Raja Rani

    (Research Fellow, School of Engineering, University of Portsmouth, University House, Winston Churchill Avenue, Portsmouth PO1 2UP, UK
    Deputy Head of General Studies Department, Military Technological College, Al Matar Street, Muscat 111, Oman)

  • Paddy Walsh

    (Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
    Senior Product Manager, Amazon, County Dublin, Burlington Road, Dublin 4, Ireland)

Abstract

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index γ ∈ [ 0 , 2 ] and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

Suggested Citation

  • Jonathan Blackledge & Derek Kearney & Marc Lamphiere & Raja Rani & Paddy Walsh, 2019. "Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction," Mathematics, MDPI, vol. 7(11), pages 1-57, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1057-:d:283546
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    References listed on IDEAS

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    1. Fama, Eugene F, 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, American Finance Association, vol. 25(2), pages 383-417, May.
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    Cited by:

    1. Florin Turcaș & Florin Cornel Dumiter & Marius Boiță, 2022. "Econophysics Techniques and Their Applications on the Stock Market," Mathematics, MDPI, vol. 10(6), pages 1-25, March.

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