Optimal Transport, Interacting Particles, and Stochastic Portfolio Theory

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The practical motivation for the questions under study in this research project comes from quantitative finance. Specifically, the project investigates the mathematics behind portfolios that outperform a standard index such as S&P 500 in the presence of volatility. The utility of such portfolios is that in that process they reduce volatility demonstrably, and thus contribute to the stability of financial markets. Although ad hoc use of such portfolios is widespread, a systematic mathematical study has been limited so far. It turns out the mathematics is related to some very modern topics in probability and geometry, especially that of the Monge-Kantorovich optimal transport maps. Successful completion of the research will not only lead to striking new mathematics in probability and information geometry, but will also imply very practical applications in modern portfolio management, with potential societal benefits of the highest level via increased financial stability. The investigator plans to study an array of problems linking the fields of optimal transport and interacting particle systems, with applications to modern portfolio theory. There are two parts to this project. The first one investigates the behavior of exponential stochastic integrals where the integrand is given by a solution of a Monge-Kantorovich optimal transport map. The study is an interesting interplay between geometry of the unit simplex and universal behavior of stochastic integrals when they are not martingales. Classical probability relies heavily on the martingale property of stochastic integrals. Here, the investigator explores a completely distinct class of behaviors that should be of independent interest. The second part studies properties of interacting continuous-time particle systems in finite and infinite dimensions. These particle systems, originally coming from stochastic portfolio theory, can be thought of as a continuous-time analogue of the discrete time exclusion process on the integer lattice. However, unlike exclusion processes, very little is known about these processes. They are conjectured to display striking phase transitions. The project aims to establish concentration inequalities and fluctuation estimates (among other properties) for such processes, which is a step towards proving more subtle behavior of these processes.