2025 Theses Doctoral

Portfolio Theory in Markets with Randomly Fluctuating Dimension and Composition

Tilva, Abhishek Kantilal; Karatzas, Ioannis

This dissertation develops a version of arbitrage theory, and of the theory of functional generation of portfolios, for piecewise semimartingale market models. These models undergo temporal changes in their dimension and composition in a random fashion.

Arbitrage theory is developed under the assumption of No Arbitrage of the First Kind (NA₁), and presented in two different parts. The first part develops the theory under the assumption of continuity and positivity of the price process between dimensional changes. Notions such as local martingale and supermartingale numéraires are discussed along with their optimality properties, and a fundamental theorem connecting all these notions is presented. Furthermore, an optional decomposition theorem is also developed. Similar results are then developed in the context of open markets.

In the second part, the assumptions of continuity and positivity of the price process are dropped, and the fundamental theorem of asset pricing is proved in the general case. A major result in the second part is the optional decomposition theorem, with whose help many results such as existence of the supermartingale numéraire, superhedging duality, and the second fundamental theorem of asset pricing, are proved.

The theory of functional generation of portfolios is developed in full generality, in a setting where capitalization processes are modeled using piecewise RCLL semimartingales. Both the additive and the multiplicative generation of portfolios are developed, and discussed in the context of rank-based generation of portfolios. Since we include jumps in the capitalization processes, we also derive a discrete-time version of the theory of functional generation. Finally, we provide an empirical analysis of some of these portfolios in real markets using a stock dataset over 40 years from two U.S. stock exchanges (NYSE and AMEX).