Estimation of hidden diffusion parameters: diffusion process integrals and stochastic volatility models.

Summary This thesis deals with the parametric estimation of the drift and diffusion coefficients of a diffusion process when we observe a functional of the trajectory, and not the trajectory itself. The first part of the thesis is devoted to the case where we observe the integral of the trajectory on consecutive time intervals. We are interested in the case where these time intervals are of fixed length and in the case where their length tends to 0. We show explicit contrasts in these two cases which lead to asymptotically Gaussian estimators, easy to implement in practice. The second part is devoted to stochastic volatility models. We consider a two-dimensional process, of which we observe only the first coordinate. This one has for diffusion coefficient the hidden diffusion whose unknown parameters are to be estimated. We construct explicit estimators of all the parameters of the hidden diffusion and determine their convergence speeds and asymptotic laws. Throughout the paper, we illustrate our results with numerical simulations on models commonly used in finance.