Soumik Pal (University of Washington)

Title: Two models of equity markets and the distribution of market weights 

Abstracts: We consider two stochastic models of equity markets as proposed by Fernholz, Karatzas, and coauthors. These are multidimensional processes where each coordinate represents the capital that a company has in the equity market. The market weights refer to the ratio of individual capital over the total capital that is present in the market. The market weight of a company represents the influence that an individual company exerts on the entire financial market. The structure and properties of market weights have long been studied in economists who are fascinated by its peculiar structure and very stable behavior.

Fernholz and Karatzas, along with coauthors, propose the 'rank-based models' and the 'volatility-stabilized market models' in order to
capture an empirically observed fact. The growth rate and volatility of a stock capital depends heavily on its current market weight. In particular, stocks with smaller market weights have higher growth rate and higher fluctuations, while the ones with larger weights are more stable. We analyze the probabilistic behavior of market weights under these models. We show that they are linked to many other areas of probability, including reflected Brownian motions, measure-valued diffusions, and the Poisson-Dirichlet point processes. We also mathematically explain a widely observed fact: a power law decay of ordered market weights in equilibrium.