Analysis of Stochstic Evolution

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.