Advances in stochastic processes with machine learning applications

Stochastic processes play a fundamental role in our attempt to model the world around us. This thesis explores several topics within this field, with a particular focus on prediction, estimation, and inference. We begin by addressing the empirical question of whether order book-based financial markets exhibit predictability, using deep learning architectures to extract information from their microstructural dynamics. Moving from discrete-time to continuous-time modeling, we then consider continuous-time autoregressive processes, developing consistent and asymptotically normal estimators for their drift parameters under both continuous and discrete observations. Finally, we present new limiting results for the empirical expected signature, a powerful non-parametric statistic of path-valued random variables, and review its application in various machine learning algorithms, demonstrating the practical benefits of our findings. A recurring theme across all three projects is the use of high-frequency data, an essential bridge between theory and practice.