Volterra process and applications in finance.

Summary This thesis is devoted to the study of Volterra processes and their use in finance. We start by recalling some properties of these processes that we will use throughout our work. The second part deals with the study of an optimal stopping problem, the valuation of an American option in a Heston-Volterra model. For some choice of parameters, this model is a so-called rough version of the well-known Heston model. We focus on the convergence of prices in a sequence of high dimensional models, approximating the original model, to prices in the Volterra limit model. In the third chapter of this work, we study the moments of Volterra polynomial processes. We propose methods to compute the moments of these processes and show that they have some properties in common with the classical polynomial diffusions. We conclude this work by focusing in the fourth chapter on more statistical problems. We address the problem of estimating the mean reversion speed parameter of a Volterra Ornstein-Uhlenbeck process. We show that our estimators, based on continuous or discrete observations of the process, are strongly consistent.