Drift Estimation For Nonparametric Diffusion Model: Nonasymptotic Approach

. We consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable. The goal is to estimate the unknown drift coefficient. We apply a locally linear smoother with a data-driven bandwidth choice. The procedure is fully adaptive and nearly optimal up to a log log factor. The results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process. 1. Introduction In this paper, we propose a procedure for adaptive estimation of the drift coefficient of a diffusion system described by the Ito equations dX t = f(X t ) dt + g(X t ) dw t ; X 0 = x 0 ; 0 t T : (1.1) Here w t is a standard Wiener process and T is the observation time. The functions f; g , entering in (1.1), which are usually referred to as drift and diffusion coefficients, are unknown. The goal is to recover the unknown drift function f from the observations X t , 0 t T . We do not discuss h..