Statistical inference of Ornstein-Uhlenbeck processes : generation of stochastic graphs, sparsity, applications in finance.

Summary The subject of this thesis is the statistical inference of multidimensional Ornstein-Uhlenbeck processes. In a first part, we introduce a model of stochastic graphs defined as binary observations of trajectories. We then show that it is possible to deduce the dynamics of the underlying trajectory from the binary observations. For this, we construct statistics from the graph and show new convergence properties in the context of a long time and high frequency observation. We also analyze the properties of stochastic graphs from the point of view of evolving networks. In a second part, we work under the assumption of complete information and continuous time and add a sparsity assumption concerning the textit{drift} parameter of the Ornstein-Uhlenbeck process. We then show sharp oracle properties of the Lasso estimator, prove a lower bound on the estimation error in the minimax sense and show asymptotic optimality properties of the Adaptive Lasso estimator. We then apply these methods to estimate the speed of return at the average of daily returns of US stocks as well as the prices of dividend futures for the EURO STOXX 50 index.