L^2-deviation for the variance estimator of a fractional diffusion
Résumé
We propose punctual estimators for the local variance $\sigma^{2}$ in pseudo-diffusions $X$ driven by a Gaussian noise. We define a kernel estimator of $\sigma^{2}$ based on the quadratic variation of $X$. The consistency and asymptotic normality are shown. Besides, a simulation study is made to assess the performance of those estimators. This study reveals, through various examples, that the estimators approximate very well the true local variance. Then we provide global estimation for the parameter $\sigma^{2}$ in this model that can be considered as a smooth perturbation of a fractional process. Thus we need to study functionals such as the Integrated Square Error (ISE) in order to obtain asymptotics laws to make global hypothesis testing. The Mean Integrated Square Error (MISE) is an usual measure of the precision of a non parametric estimator. A precise asymptotic expansion of the MISE gives optimal bandwith choice. These results with those concerning the ISE lead to contiguity tests for $\sigma$. Estimating the integral of the square of the second derivative of $\sigma^{2}$ also leads to a test of linearity. The proofs are simplified by using the Central Limit Theorem for non-linear functionals that belong to Itô-Wiener's Chaos, of Peccati-Nualart-Tudor.