Non-parametric drift estimation for diffusions from noisy data

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 = kδ for k from 1 to N, we have at disposal noisy data of the sample path, Y_kδ = X_kδ+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 = pδ is small whereas Nδ 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.