Asymptotic study of stochastic algorithms and computation of Parisian options prices.

Summary This thesis deals with two independent subjects. The first part is devoted to the study of stochastic algorithms. In a first introductory chapter, I present the algorithm of [55] in a parallel with Newton's algorithm for deterministic optimization. These few reminders then allow me to introduce the randomly truncated stochastic algorithms of [21] which are at the heart of this thesis. The first study of this algorithm concerns its almost sure convergence which is sometimes established under rather changing assumptions. This first chapter is an opportunity to clarify the assumptions of the almost sure convergence and to present a simplified proof. In the second chapter, we continue the study of this algorithm by focusing this time on its speed of convergence. More precisely, we consider a moving average version of this algorithm and prove a central limit theorem for this variant. The third chapter is devoted to two applications of these algorithms to finance: the first example presents a method for calibrating the correlation for multidimensional market models while the second example continues the work of [7] by improving its results. The second part of this thesis focuses on the valuation of Parisian options based on the work of Chesney, Jeanblanc-Picqué, and Yor [23]. The valuation method is based on obtaining closed formulas for the Laplace transforms of prices with respect to maturity. We establish these formulas for single and double barrier Parisian options. We then study a numerical inversion method for these transforms. We establish a result on the accuracy of this very efficient numerical method. On this occasion, we also prove results related to the regularity of prices and the existence of a density with respect to the Lebesgues measure for Parisian times.