TY - JOUR

T1 - Econophysics and fractional calculus

T2 - Einstein's evolution equation, the fractal market hypothesis, trend analysis and future price prediction

AU - Blackledge, Jonathan

AU - Kearney, Derek

AU - Lamphiere, Marc

AU - Rani, Raja

AU - Walsh, Paddy

N1 - Publisher Copyright:
© 2019 by the authors.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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.

AB - 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.

KW - Diffusion equation

KW - Efficient market hypothesis

KW - Einstein's evolution equation

KW - Evolutionary computing

KW - Financial time series analysis

KW - Fractal market hypothesis

KW - Fractional diffusion equation

KW - Kolmogorov-Feller equation

KW - Random market hypothesis

KW - Self-affine stochastic fields

UR - http://www.scopus.com/inward/record.url?scp=85075375118&partnerID=8YFLogxK

U2 - 10.3390/math7111057

DO - 10.3390/math7111057

M3 - Article

SN - 2227-7390

VL - 7

JO - Mathematics

JF - Mathematics

IS - 11

M1 - 1057

ER -