Non-parametric drift estimation for diffusions from noisy data

Schmisser Emeline

Statistics & Risk Modeling, 2011, vol. 28, issue 2, 119-150

Abstract: Consider a diffusion process (Xt)t ≥ 0, with unknown drift b(x) and diffusion coefficient σ(x), which is strictly stationary, ergodic and β-mixing. At discrete times tk = kδ for k from 1 to N, we have at disposal noisy data of the sample path, Ykδ = Xkδ+εk. The random variables (εk) are i.i.d., centred and independent of (Xt). 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.

Keywords: drift; model selection; noisy data; non-parametric estimation; stationary distribution (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1524/stnd.2011.1063

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