Signatures in machine learning and finance

Developed to give meaning to differential equations driven by rough signals, rough path theory has opened in recent years a new approach to tackle certain problems in other fields such as mathematical finance and machine learning. This is due to certain algebraic and analytical properties of an object called the rough path signature. This thesis aims to make a contribution in this direction by introducing signature-based methods to study various problems arising in finance and machine learning. In the context of finance, we consider two problems: the pricing and hedging of financial derivatives, and the optimal execution of financial securities. We propose numerical methods based on signatures to solve both problems. Finally, in the context of machine learning we begin by introducing a new neural network architecture called the deep signature model, based on the deep signature transform. The thesis concludes with a practical application of the signature transform on a real-world problem, where we aim to make predictions from psychiatric data using signatures.