Causal Optimal Transport in Mathematical Finance

The main aim of the field of Mathematical Finance is to provide accurate and quantitative answers to such questions as: How risky is an investment in a financial market? How can we decrease that risk? How should we invest in a financial market in order to achieve a certain goal? What is a fair price for a financial instrument? After the financial crash of 2008 these and related questions have become all the more relevant, and not just for financial institutions but also for regulators and society as a whole. Much of the development in the field of Mathematical Finance is model-based. This means that in order to analyse the aforementioned questions we first must set up a mathematical model of the financial market. The aim of this project is to develop a better understanding of the impact that model selection has when we answer the above questions. Our driving idea is to consider the whole space of financial models and study its geometric structure. This will allow us to assess when two models are close to each other, or how one model should be infinitesimally perturbed in order to fit new data, etc. The ultimate goal is that, once our geometric understanding has been refined, we should be able to estimate the degree of certainty we have when we use a given model to provide an answer to a question concerning financial markets.