Analysis of Stochstic Evolution

TL;DR

This paper investigates the tail behaviors of distributions generated by geometric Brownian motion and links these findings to the widespread power-law phenomena observed in economics and other fields.

Contribution

It provides a detailed analysis of the upper and lower tails of GBM distributions and connects these results to the emergence of power-law behavior from heterogeneous agents.

Findings

Analysis of tail behaviors of GBM distributions

Correlation between tail properties and power-law emergence

Explanation of power-law universality across disciplines

Abstract

Many studies in Economics and other disciplines have been reporting distributions following power-law behavior (i.e distributions of incomes (Pareto's law), city sizes (Zipf's law), frequencies of words in long sequences of text etc.)[1, 6, 7]. This widespread observed regularity has been explained in many ways: generalized Lotka-Volterra (GLV) equations, self-organized criticality and highly optimized tolerance [2,3,4]. The evolution of the phenomena exhibiting power-law behavior is often considered to involve a varying, but size independent, proportional growth rate, which mathematically can be modeled by geometric Brownian motion (GBM) $dX_t = r_t X_t dt + \alpha X_t W_t$ where $W_t$ is white noise or the increment of a Wiener process. It is the primary purpose of this article to study both the upper tail and lower tail of the distribution following the geometric Brownian motion and to correlate this study with recent results showing the emergence of power-law behavior from heterogeneous interacting agents [5]. The result is the explanation for the appearance of similar properties across a wide range of applications.