Limit theorems for Gaussian fields via Chaos Expansions and Applications

TL;DR

This thesis develops limit theorems for Gaussian fields using chaos expansions, combining Malliavin calculus and Stein's method, with applications in cosmology and finance.

Contribution

It introduces new probabilistic approximation techniques for Gaussian fields and applies them to models in cosmology and finance, integrating large deviations theory.

Findings

Quantitative CLTs for non-linear functionals of random hyperspherical harmonics

Analysis of fractional Ornstein-Uhlenbeck process in rough volatility models

Application of large deviations to stochastic volatility models

Abstract

In this PhD thesis, we apply a combination of Malliavin calculus and Stein's method in the framework of probability approximations. The specific problems we tackle with these methods are motivated by probabilistic models in cosmology (Part I: Quantitative CLTs for non linear functionals of random hyperspherical harmonics) and finance (Part II: The fractional Ornstein-Uhlenbeck process in rough volatility modelling). In this second part we also apply techniques from Large Deviations theory (Section: Short-time asymptotics for non self-similar stochastic volatility models).