In this talk we start with an overview of the modern theory of portfolios, based on Stochastic Analysis. We introduce the notion of relative arbitrage and provide simple, easy-to-test criteria for the existence of such arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be "sufficiently large". We show that conditions satisfying these criteria are manifestly present in the US equity market, and construct explicit portfolios under these conditions. One such condition, market diversity, emerges when the volatility structure is bounded. We then construct examples of abstract markets in which the criteria hold. We study in some detail a specific example of a non-diverse abstract market which is volatility-stabilized, in that the return from the market portfolio has constant drift and variance rates, while the smallest stocks are assigned the largest volatilities and individual stocks fluctuate widely. An interesting probabilistic structure emerges in which time changes, Bessel processes, and the asymptotic theory for planar Brownian motion, play crucial roles. Several open questions are raised for further study. (Joint work with E. Robert Fernholz.) See also
             http://www.math.columbia.edu/~ik/FernKarSPT.pdf
for a paper of interest.