tochastic processes are used to model quantities that exhibit random fluctuations over time, such as stock prices, temperatures, wind speeds, etc. Lévy processes make up a popular class of models due to their theoretical properties and applications in e.g. finance and physics.
From a theoretical point of view these processes evolve in continuous time but in practice only a finite number of observations are available. Understanding the implications of this discretization is essential and in many cases it requires knowledge of the local properties of the process. The first paper in this dissertation is concerned with a specific discretization scheme for positive self-similar Markov processes. To describe this we rely on knowledge about small-time fluctuations of Lévy processes. In another paper we examine the local properties of diffusions. At a fixed time point such a process behaves locally as a scaled Brownian motion and we prove a similar result for the small-time fluctuations at the supremum. The theory which describes a univariate Lévy process before and after its supremum is well-known and relies on the notion of a Lévy process conditioned to stay positive or negative. In a third paper we extend this to the multivariate setting, constructing the law of a Lévy process conditioned to stay in a half-space. This is related to splitting the process at its directional supremum and we further conjecture how it can be used to describe the local behavior of the process when it is farthest from the origin.
One of the big challenges in modern statistics is dealing with high-dimensional data. Classical models are faced with an increased risk of overfitting, large computational cost and low interpretability. The concept of sparsity addresses these issues by taking advantage of lower-dimensional structures in the data. The use of graphical models is one way of promoting sparsity and has recently been introduced in multivariate extreme value theory. The final paper in this dissertation introduces graphical models in the context of Lévy processes. To do this we exploit a subtle connection to extremes.
