Study of some statistical estimation problems in finance.

Summary This thesis deals with several statistical finance problems and consists of four parts. In the first part, we study the question of estimating the persistence of volatility from discrete observations of a diffusion model over an interval [0,T], where T is a fixed objective time. For this purpose, we introduce a fractional Brownian motion of Hurst index H in the volatility dynamics. We construct an estimation procedure of the parameter H from the high frequency data of the diffusion. We show that the precision of our estimator is n^{-1/(4H+2)}, where n is the observation frequency and we prove its optimality in the minimax sense. These theoretical considerations are followed by a numerical study on simulated and financial data. The second part of the thesis deals with the problem of microstructure noise. For this, we consider observations at frequency n and with rounding error α_n tending to zero, of a diffusion model over an interval [0,T], where T is a fixed objective time. In this framework, we propose estimators of the integrated volatility of the asset whose precision is shown to be max(α_n, n^{-1/2}). We also obtain central limit theorems in the case of homogeneous diffusions. This theoretical study is also followed by a numerical study on simulated and financial data. In the third part of this thesis, we establish a simple characterization of Besov spaces and we use it to prove new regularity properties for some stochastic processes. This part may seem disconnected from the problems of statistical finance but it has been inspiring for part 4 of the thesis. In the last part of the thesis, a new microstructure noise index is constructed and studied on financial data. This index, whose calculation is based on the p-variations of the considered asset at different time scales, can be interpreted in terms of Besov spaces. Compared to other indices, it seems to have several advantages. In particular, it allows to highlight original phenomena such as a certain form of additional regularity in the finest scales. It is shown that these phenomena can be partially reproduced by additive microstructure noise or diffusion models with rounding error. Nevertheless, a faithful reproduction seems to require either a combination of two forms of error or a sophisticated form of rounding error.