Optimization Problems in Model-Free Stochastic Portfolio Theory and Sequential Testing Games

Abstract

The principal focus of this dissertation is on problems in Mathematical Finance whose solutions (i) help inform optimal individual actions, or (ii) lend understanding to the downstream behaviour of complex systems. The first part of this thesis addresses questions in Stochastic Portfolio Theory, a field that takes a descriptive approach to the study of financial markets and portfolio selection. The direction (i) incentivizes the study of a model-free non-parametric optimization problem related to the important infinite dimensional family of functionally generated portfolios which has desirable theoretical properties and can be implemented numerically. Insights from this work motivate the development of the statistical and computational theory of Convex PCA, a dimension reduction methodology for data living in a convex subset of a Hilbert space. In line with (ii), the applications generate new insights about the principal features of financial markets. The second part of this thesis begins by introducing and solving a soft classification version of a Bayesian sequential testing problem whose history dates back to Wald. This formulation is then embedded in a game setting where a continuum of agents interact and influence each other’s decisions. The study of this game presents the first treatment in the literature of a tractable mean field game with information filtering, optimal stopping, and a common unobserved noise. In the spirit of (i) and (ii), an investigation of the solution reveals the optimal decisions agents make and the nature of the equilibria that arise.