Non-parametric estimation of the diffusion coefficient from noisy data

Emeline Schmisser ()

Statistical Inference for Stochastic Processes, 2012, vol. 15, issue 3, 193-223

Abstract: We consider a diffusion process (X t ) t ≥ 0 , with drift b(x) and diffusion coefficient σ(x). At discrete times t k = k δ 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 (X t ). The process (X t ) t ≥ 0 is assumed to be strictly stationary, β-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 Δ = p δ is small whereas M δ is large. Then, the diffusion coefficient σ 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. Copyright Springer Science+Business Media Dordrecht 2012

Keywords: Diffusion coefficient; Model selection; Noisy data; Non-parametric estimation; Stationary distribution; Primary 62G08; Secondary 62M05 (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11203-012-9072-8

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