Analysis of Stochstic Evolution
Francesco Vallone

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) where 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…
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Taxonomy
TopicsComplex Systems and Time Series Analysis · Evolutionary Game Theory and Cooperation · Theoretical and Computational Physics
