Estimation of Local Smoothness Coefficients for Continuous Time Processes

We consider processes that satisfied a local Hölder condition with coefficient ₀. According to the sampling times of observations given by i_n with i=0,...,n–1, we study two general classes of estimators for ₀. Their almost sure rates of convergence depend on asymptotic independence of the observed processes, on _n and eventually on an extra parameter ₀. Since this last parameter is in general unknown, we construct a family of preliminary estimators for ₀ with their rates of almost sure convergence. Finally we present some numerical simulations in order to compare the behaviour of our various estimators.