Max-Stable Processes, Measure Transport & Conditional Sampling

Abstract

The modeling of extremes, known as extreme value theory (EVT), aims to understand events characterized by extreme deviations from the mean of a probability distribution. These events are significant in fields such as finance, environmental science, engineering, and insurance. EVT aims to predict the occurrence and impact of these events, which often have severe consequences. Applications of EVT include modeling extreme market movements in finance, natural disasters in environmental sciences, structural reliability in engineering, and catastrophic event risk management in insurance. Conditional sampling and simulation methods, such as normalizing flows and measure transport, are crucial for estimating extremes at un-monitored sites or under specific conditions, thereby improving our understanding and risk management strategies. The goal of this thesis is to make significant contributions to both extreme value theory and measure transport, as well as to establish a link between the two. First, we develop new Markov chain Monte Carlo algorithms for conditional sampling of max-stable processes. Next, we create models that incorporate physical laws, encoded by partial differential equations, to extend max-stable processes into regions without observations. Third, we design specialized transport map frameworks for distributions with bounded support, enabling accurate and efficient sampling and inference. Finally, we use transport maps parameterized by neural networks to learn and condition the distributions of shortest path statistics in polymer systems, accelerating the prediction of microstructural evolution under various conditions.