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

Jonathan Blackledge, Derek Kearney, Marc Lamphiere, Raja Rani and Paddy Walsh
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Jonathan Blackledge: Stokes Professor, Science Foundation Ireland, Three Park Place, Dublin 2, Ireland
Derek Kearney: Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
Marc Lamphiere: Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland
Raja Rani: Research Fellow, School of Engineering, University of Portsmouth, University House, Winston Churchill Avenue, Portsmouth PO1 2UP, UK
Paddy Walsh: Dublin Energy Laboratory, Technological University Dublin, Kevin Street, Dublin 8, Ireland

Mathematics, 2019, vol. 7, issue 11, 1-57

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.

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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