Drift estimation on non compact support for diffusion models

Abstract : In this paper, we study non parametric drift estimation for an ergodic and β-mixing diffusion process from discrete observation of the sample path. The drift is estimated on a set A using an approximate regression equation by a least squares contrast which is minimized over a finite dimensional subspace Sm of L 2 (A, dx). This yields a collection of estimators indexed by the dimension of the projection space. The novelty here is that, contrary to previous works, the set A is general and may be non compact and the diffusion coefficient may be unbounded. This induces a restricted set for the possible dimensions of the projection spaces. Under mild assumptions, risk bounds of a L 2-risk are provided where new variance terms are exhibited. A data-driven selection procedure by penalization is proposed where the dimension of the projection space is chosen within a random set contrary to usual selection procedures. Risk bounds are obtained showing that the resulting estimator is adaptive in the sense that its risk achieves automatically the bias variance compromise. The estimation method is illustrated on simulated data for several diffusion models.
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Submitted on : Thursday, November 8, 2018 - 3:05:03 PM
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Fabienne Comte, Valentine Genon-Catalot. Drift estimation on non compact support for diffusion models. 2018. ⟨hal-01916503⟩

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