Non-parametric estimation of the diffusion coefficient from noisy data

Abstract : Abstract We consider a diffusion process \left(X_{t}\right)_{t\geq0}, with drift b(x) and diffusion coefficient \sigma(x). At discrete times t_{k}=k\delta for k from 1 to M, we observe 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). The process \left(X_{t}\right)_{t\geq0} is assumed to be strictly stationary, \beta-mixing and ergodic. 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, the diffusion coefficient \sigma^{2} is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.
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Statistical Inference for Stochastic Processes, Springer Verlag, 2012, 15 (3), pp 193-223. <10.1007/s11203-012-9072-8>
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Emeline Schmisser. Non-parametric estimation of the diffusion coefficient from noisy data. Statistical Inference for Stochastic Processes, Springer Verlag, 2012, 15 (3), pp 193-223. <10.1007/s11203-012-9072-8>. <hal-00443993>

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