Portfolio theory, optimal transport and information geometry

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Quantitative investment strategies play a decisive role in modern financial markets. They determine how financial institutions deal with their frequent tradings and how pension funds manage our savings. These strategies are based on theoretical and empirical models of financial markets. Many strategies are highly model-specific: if the model is wrong, or when market conditions change, disastrous events may happen. Thus, there is a strong need to study robust investment strategies, i.e., strategies whose success do not depend on the assumptions of a specific model. This research program develops robust portfolio theory using novel tools from probability and geometry. Simultaneously, we advance the mathematical theories which have found important applications in data science. The financial ideas come from stochastic portfolio theory and universal portfolio theory. These theories provide robust investment strategies based on persistent features of financial markets and ways to combine them; we propose to combine these approach under more realistic settings. Mathematically, these strategies are related to optimal transport which is about assigning market scenarios to asset allocations in an overall cost-efficient way. Geometry enters the picture if we think of the evolving market as a point traveling in a high dimensional space. We endow the space with a suitable geometry such that the directions and magnitudes of market movements have direct impacts on the portfolio. This geometry can be studied using optimal transport and information geometry. We propose to study the empirical properties of the market from this geometric viewpoint. By adopting a multi-disciplinary approach, we hope to develop novel mathematical theories with practical applications including algorithms and software packages. In particular, we aim to generalize the classical Wasserstein geometry by studying new cost functions. The outcomes of the research program will provide robust tools to manage portfolios and risks related to market concentration and volatility. They are also expected to improve our understanding about optimal transport and information geometry whose connections have started to attract a lot of attention. The mathematical results and algorithms developed are expected to be useful beyond quantitative trading and will lead to further theoretical development and applications in statistics and machine learning.