Théorème de Girsanov et Applications

Abstract

This thesis explores Girsanov's Theorem and its applications in stochastic processes, focusing on the Ornstein-Uhlenbeck (O-U) process and Itô-Lévy processes. The study begins with foundational concepts of stochastic processes, including Brownian motion, martingales, and Itô calculus. It then rigorously presents Girsanov’s Theorem, which allows changing the probability measure to transform a stochastic process with drift into a driftless one (or vice versa). The applications demonstrate how Girsanov’s Theorem simplifies the analysis of O- U processes (used in finance and physics) and Itô-Lévy processes (incorporating jumps). Simulations in R illustrate the theoretical results, highlighting the theorem’s utility in financial modeling and risk-neutral pricing.