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Open AccessFeature PaperArticle

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

by Jonathan Blackledge 1,2,3,4,5,6,*, Derek Kearney 6,7, Marc Lamphiere 6,8,9, Raja Rani 10,11 and Paddy Walsh 6,12
1
Stokes Professor, Science Foundation Ireland, Three Park Place, Dublin 2, Ireland
2
Honorary Professor, School of Electrical and Electronic Engineering, Technological University, Kevin Street, Dublin 8, Ireland
3
Professor Extraordinaire, Department of Computer Science, University of Western Cape, Bellville 7535, Cape Town, South Africa
4
Honorary Professor, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Westville Campus, University Road, Westville 3630, Durban, South Africa
5
Visiting Professor, Faculty of Arts, Science and Technology, University of Wales (Wrexham Glyndwr University), Mold Road, Wrexham LL11 2AW, UK
6
Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
7
Lecturer, School of Electrical and Electronic Engineering, Technological University Dublin, Kevin Street, Dublin 8, Ireland
8
Research Associate, School of Electrical and Electronic Engineering, Technological University Dublin, Kevin Street, Dublin 8, Ireland
9
Country Director, The Natural Power Consultants (Ireland) Limited, Beacon Court Sandyford, Dublin 18, Ireland
10
Research Fellow, School of Engineering, University of Portsmouth, University House, Winston Churchill Avenue, Portsmouth PO1 2UP, UK
11
Deputy Head of General Studies Department, Military Technological College, Al Matar Street, Muscat 111, Oman
12
Senior Product Manager, Amazon, County Dublin, Burlington Road, Dublin 4, Ireland
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1057; https://doi.org/10.3390/math7111057
Received: 6 September 2019 / Revised: 18 October 2019 / Accepted: 19 October 2019 / Published: 4 November 2019
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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. View Full-Text
Keywords: Einstein’s evolution equation; Kolmogorov–Feller equation; diffusion equation; fractional diffusion equation; self-affine stochastic fields; random market hypothesis; efficient market hypothesis; fractal market hypothesis; financial time series analysis; evolutionary computing Einstein’s evolution equation; Kolmogorov–Feller equation; diffusion equation; fractional diffusion equation; self-affine stochastic fields; random market hypothesis; efficient market hypothesis; fractal market hypothesis; financial time series analysis; evolutionary computing
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Blackledge, J.; Kearney, D.; Lamphiere, M.; Rani, R.; Walsh, P. Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction. Mathematics 2019, 7, 1057.

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