On some Inferences of Lévy Distribution

The levy distribution is one of the three distributions that has probability density function in simple closed form. This distribution is used in modeling stock prices. In this paper, we present some properties of this distribution. Based on the basic properties some characterizations of this distribution are given.

Simulations of Lévy walk

We simulate stable distributions to study the ideal movement pattern for the spread of a virus using autonomous carrier. We observe Lévy walks to be the most ideal way to spread and further study how the parameters in Lévy distribution affects the spread.

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

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.