The topic motivating this dissertation is functionally generated portfolios and their capacity to deliver relative arbitrage, an aspect of stochastic portfolio theory (SPT). The aim is to relax some of the common assumptions of SPT and explore the performance of functionally generated portfolios in this more general setting, with an eye towards arbitrage. In particular, the assumption of a constant number of companies in the market model is relaxed, as well as the assumption that all changes in capitalizations are passed on as returns to investors through the stochastic integral.

On the way to these goals, the notions of a piecewise semimartingale taking values in [special characters omitted] and piecewise stochastic integration are developed as useful mathematical tools. Many properties of the stochastic integral with respect to an [special characters omitted]-valued semimartingale are shown to extend to this setting. For example, the following fundamental theorems of asset pricing carry over: "No free lunch with vanishing risk" is equivalent to the existence of an equivalent sigma-martingale measure and "No arbitrage of the first kind" is equivalent to the existence of an equivalent local martingale deflator for the integrator.

An important idea of SPT is the notion of diversity of equity markets, meaning that relative capital does not become arbitrarily concentrated in a single company. In [38] Robert Fernholz observed an inconsistency between the normative assumption of existence of an equivalent local martingale measure (ELMM) for the price process and the empirical reality of diversity in equity markets. Here an alternative model is examined, in which diversity is maintained not through smaller returns of the largest company, but via a type of antitrust regulation that is compatible with ELMMs. The regulatory procedure breaks up companies that become too large, while assuming that total capital in the market is conserved. In this case and in several other examples where the price process is a piecewise Itô process, straightforward functionally generated relative arbitrage is found to be less readily available than in n-dimensional Itô process models.