Non-Linear Diffusion and Power Law Properties of Heterogeneous Systems: Application to Financial Time Series.

Miguel A Fuentes

doi:10.3390/e20090649

Abstract

In this work, we show that it is possible to obtain important ubiquitous physical characteristics when an aggregation of many systems is taken into account. We discuss the possibility of obtaining not only an anomalous diffusion process, but also a Non-Linear diffusion equation, that leads to a probability distribution, when using a set of non-Markovian processes. This probability distribution shows a power law behavior in the structure of its tails. It also reflects the anomalous transport characteristics of the ensemble of particles. This ubiquitous behavior, with a power law in the diffusive transport and the structure of the probability distribution, is related to a fast fluctuating phenomenon presented in the noise parameter. We discuss all the previous results using a financial time series example.

Highlights

In recent years, many papers have been written focusing on the study of anomalous collective motions and particularities on probability distributions
We could say that the ubiquitous characteristics that in principle are present in the systems under study have two remarkable properties: power law behavior in the structure of its distribution and dynamic characteristics of a system of many particles with anomalous diffusion
Let us start the presentation by studying the microscopic dynamics of a Brownian particle, where dissipation is described by a memory kernel γ(t)

Summary

Introduction

Many papers have been written focusing on the study of anomalous collective motions and particularities on probability distributions. It is interesting to note, as we mentioned before, that the characteristics associated with the collective movement of some of these systems, where the second moment, i.e., < x (t)2 >= tα , being α = 1 the value for the exponent where the system behaves in a normal way, is the typical quantity to be studied [6]. In this contribution, we will describe particular features at the microscopic level of the system, and how they will impact on the macroscopic characteristics of such behaviors, focusing on a financial time series example.

Microscopic Dynamics

Power Law Behavior in the Movement of Ensemble of Particles

A Marginalization of Weakly Coupled Systems

Application to the Financial Time Series

Final Remarks