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Statistics & Risk Modeling

with Applications in Finance and Insurance

Editor-in-Chief: Stelzer, Robert

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Non-parametric drift estimation for diffusions from noisy data

Emeline Schmisser*

* Correspondence address: Université Paris Descartes, Laboratoire MAP 5, 45 rue des Saints Pères, 75270 Paris Cedex 06, Frankreich,

Citation Information: Statistics & Decisions International mathematical journal for stochastic methods and models. Volume 28, Issue 2, Pages 119–150, ISSN (Print) 0721-2631, DOI: 10.1524/stnd.2011.1063, May 2011

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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

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Emeline Schmisser
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