Economics and Finance: q-Statistical Stylized Features Galore

Constantino Tsallis

doi:10.3390/e19090457

Abstract

The Boltzmann–Gibbs (BG) entropy and its associated statistical mechanics were generalized, three decades ago, on the basis of the nonadditive entropy S q ( q ∈ R ), which recovers the BG entropy in the q → 1 limit. The optimization of S q under appropriate simple constraints straightforwardly yields the so-called q-exponential and q-Gaussian distributions, respectively generalizing the exponential and Gaussian ones, recovered for q = 1 . These generalized functions ubiquitously emerge in complex systems, especially as economic and financial stylized features. These include price returns and volumes distributions, inter-occurrence times, characterization of wealth distributions and associated inequalities, among others. Here, we briefly review the basic concepts of this q-statistical generalization and focus on its rapidly growing applications in economics and finance.

Highlights

Exponential and Gaussian functions ubiquitously emerge within linear theories in mathematics, physics, economics and elsewhere
In the realm of social systems, on, we focus on economics and financial theory [102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118]
Wealth inequality within a given country is a classical and most important matter, which can be characterized within q-statistics as shown in [105]: see Figures 12 and 13

Summary

Introduction

Exponential and Gaussian functions ubiquitously emerge within linear theories in mathematics, physics, economics and elsewhere. The solution is the well-known Gaussian distribution: e−x /2Dt. Let us consider the following entropic functional: SBG = −k with the constraint: Z dx p(x) ln p(x) (k > 0). Basic cases connect SBG with the solutions of the linear Equations (1) and (3). The important connection between the power-law nonlinear diffusion (12) and the entropy Sq described here below was first established by Plastino and Plastino in [11], where they considered a more general evolution equation that reduces to (12) in the particular case of vanishing drift (i.e., F(x) = 0, ∀x). Let us focus on the entropic functional Sq upon which nonextensive statistical mechanics is based. It is defined as follows: Sq ≡ k. In the realm of social systems, on, we focus on economics and financial theory [102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118]

Prices and Volumes

Volatilities

Inter-Occurrence Times

Wealth

Conclusions and Perspectives