Nonparametric estimation of the diffusion coefficient of an ergodic diffusion process on non-compact supports under Osgood's conditions

In this paper, we study the nonparametric estimation of the squared diffusion coefficient of an ergodic diffusion process on non-compact supports, when the two coefficients of the stochastic differential equation belong to a space larger than the space of Hölder continuous functions. The estimators are constructed by projection onto finite-dimensional spaces, thereby avoiding truncation of the dimension. We establish risk bounds of non-adaptive estimators and explicit rates of convergence for bounded and unbounded diffusion coefficients. A model selection procedure is performed, followed by the derivation of risk bounds using Talagrand's inequality. The theoretical results are completed with a numerical study over simulated data.