Topics in Stochastic Portfolio Theory

This thesis generalizes stochastic portfolio theory in two different aspects. The first part demonstrates the functional generation of portfolios in a pathwise way. This notion of functional generation of portfolios was first introduced by E.R. Fernholz, to construct a variety of portfolios solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf developed recently another approach to the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. Both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions, possibly less smooth than twice-differentiable, involve the current market weights, as well as additional bounded-variation functionals of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible to analyze, and to improved conditions for outperforming the market portfolio over suitable time-horizons. The second part develops portfolio theory in open markets. An open market is a subset of the entire equity m