Nonparametric drift estimation for diffusions from noisy data

Abstract : Let us consider a diffusion process \left(X_{t}\right)_{t\geq0}, with drift b(x) and diffusion coefficient \sigma(x). This process is assumed to be strictly stationnary, \beta-mixing and ergodic. At discrete times t_{k}=k\delta for k from 1 to M, we have at disposal noisy data of the sample path, Y_{k\delta}=X_{k\delta}+\varepsilon_{k}. The random variables \left(\varepsilon_{k}\right) are i.i.d, centred and independent of \left(X_{t}\right). 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 \Delta=p\delta is small whereas M\delta is large. Then, we estimate the drift function b in a compact set A in a nonparametric way by a penalized least squares approach. We obtain a bound for the risk of the resulting adaptive estimator. We also provide several examples of diffusions satisfying our assumptions and realise various simulations. Our simulation results illustrate the theoretical properties of our estimators.
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Statistics & Decisions, 2011, 28 (2), pp.119-150. <10.1524/stnd.2011.1063>
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Contributeur : Emeline Schmisser <>
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Emeline Schmisser. Nonparametric drift estimation for diffusions from noisy data. Statistics & Decisions, 2011, 28 (2), pp.119-150. <10.1524/stnd.2011.1063>. <hal-00419954>

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