Parisian times, Bessel processes and Poisson-Dirichlet random variables

In this thesis, we study the first hitting time and Parisian time of Brownian motion and squared Bessel process, as well as the exact simulation algorithm of the two-parameter Poisson-Dirichlet distribution. Let the underlying process be a reflected Brownian motion with drift moving on a finite collection of rays. Using a recursive method, we derive the Laplace transform of the first hitting time of the underlying process. This generalises the well-known result about the first hitting time of a Walsh Brownian motion on spider. We also invert the Laplace transform explicitly using two different methods, and obtain the density and distribution functions of the first hitting time. Then we consider the Parisian time of the underlying process, which is defined as the first exceeding time of the excursion time length. Using the same recursive method, we derive the Laplace transform of the Parisian time. The exact simulation algorithm for the Parisian time is also proposed. Next, we extend the result to the Parisian time of a squared Bessel process with a linear excursion boundary. Based on a variation of the Azéma martingale, we obtain the distributional properties of the Parisian time, and design the algorithm for sampling from the Parisian time. Finally, as an extension of the simulation of the Parisian time, we propose two exact simulation algorithms for sampling from the two-parameter Poisson-Dirichlet distribution.