Dr. Tomoyuki Ichiba (PSTAT/CRFMS Postdoctoral Fellow, UCSB)

Title: Hybrid Atlas Models (Part I)

We study Atlas type models of equity markets with local characteristics that depend on both name and rank, and in ways that explain a stability of empirical capital distribution. This study involves rankings of continuous semimartingales and the local times of the differences between adjacent processes. By a comparison argument with Bessel processes we derive a representation of the market in terms of the reflected Brownian motion that yields invariant distribution of market capital shares under some assumptions. The class of resulting expected capital distribution curves seem rich enough to explain the empirical curves. We also discuss various portfolios including universal portfolios.

Topics: (sub)martingale problems, rankings of continuous semimartingales, attainability of diffusions, collisions of Brownian particles, reflected Brownian motions, stochastic stability, average occupation time, capital distributions, stochastic portfolio theory.