Non-parametric drift estimation for diffusions from noisy data

Abstract

Abstract Consider a diffusion process (X t ) t ≥ 0, with unknown drift b(x) and diffusion coefficient σ(x), which is strictly stationary, ergodic and β-mixing. At discrete times t k = k δ for k from 1 to N, we have at disposal noisy data of the sample path, Y k δ = X k δ +ε k . The random variables (ε k ) are i.i.d., centred and independent of (X t ). In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that Δ = p δ is small whereas N δ is large. Then, the drift function b is estimated in a compact set A in a non-parametric way using a penalized least squares approach. We obtain a bound for the risk of the resulting adaptive estimator. Examples of diffusions satisfying our assumptions are given and numerical simulation results illustrate the theoretical properties of our estimators.