This thesis presents a study of the signature of a path, explores the concept of expected signatures and their relationship to the laws of random variables. The signature of a path is defined as the collection of iterated integrals of the path, and it has several desirable properties that make it a useful tool for analyzing and understanding high-dimensional data. In particular, we focus on the properties of invariance under time reparameterizations, the shuffle product, and Chen’s identity, as well as the time-reversal and log signature. For an in-depth examination of the connection between expected signatures and stochastic processes, this thesis delves into the concepts of the universal locally m-convex algebra, group-like elements, and representations of the universal topological algebra. The theoretical results are applied to the Heston Model and the results of the simulation are presented. Simulations are used to analyze the results and draw conclusions about the usefulness of expected signatures in gaining an overview of the stochastic process dynamics. Overall, the signature of a path is a powerful tool for analyzing and understanding high-dimensional paths, with potential applications in a wide range of fields.      «

This thesis presents a study of the signature of a path, explores the concept of expected signatures and their relationship to the laws of random variables. The signature of a path is defined as the collection of iterated integrals of the path, and it has several desirable properties that make it a useful tool for analyzing and understanding high-dimensional data. In particular, we focus on the properties of invariance under time reparameterizations, the shuffle product, and Chen’s identity...     »