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

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.

Carbon Futures Trading and Short-Term Price Prediction: An Analysis Using the Fractal Market Hypothesis and Evolutionary Computing

“…In this section, we discuss the background to Einstein's Evolution Equation, the Random, Efficient, and Fractal Market Hypotheses and the approach to the development of a carbon market future trend indicator. This is a summary only, as this work is detailed more fully in an associated publication by the authors of Reference [36]. Interested readers should refer to this paper for a fuller derivation of the underlying hypothesis and its application to financial markets in general.…”

“…For γ = 2, a Gaussian PDF is obtained; for γ = 1, the PDF is a Cauchy distribution, and, with γ = 0, the PDF is a delta function. Specifically, for r ∈ R 1 , it can be shown that, for γ = 1 [36],…”

“…The Lyapunov exponent can be derived from EEE [36], and, for a continuous time series x(t) with an initial value x 0 , the (leading) Lyapunov Exponent λ can be shown to be given by…”

“…For a discrete time series which is based on some iterative process in which an error occurs denoted by (t n ), n = 1, 2, ..., N and is characterised by exponential growth exp(λτ) where τ is the sampling rate, the Lyapunov Exponent can be written in the form [36]…”

Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries

“…The use of semi-derivatives and integrals in the mass and heat transfer become an important instant in the field of fractional calculus due to employing the mathematical definitions into physical phenomena [12,13]. In the last decade, using fractional operators which explain the events, situations or modes between two different stages or the phenomena with memory provide more accurate models in many branches of science and engineering including chemistry, biology, biomedical devices, nanotechnology, diffusion, diffraction and economics [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. In References [25][26][27][28][29][30][31], the modeling and comparison of the countries and trends in the sense of economics and its parameters are implemented.…”