Equivalence for nonparametric drift estimation of a diffusion process and its Euler scheme
2013
Laredo , Catherine (INRA , Jouy-En-Josas (France). UR 0341 Unité de recherche Mathématiques et Informatique Appliquées) | Genon-Catalot , Valentine (Ministere de l'Agriculture et de la PecheUniversité Paris Descartes (Paris 5), ParisParis(France).)
At first consider a stationary and ergodic stochastic process which fulfills a nonparametric diffusion model driven by a Brownian motion. Our aim is the estimation of the unknown drift and volatility functions using a discrete high-frequency sample. Motivated by Bandi and Phillips, [1], we use Nadaraya-Watson like estimators and extend their results to bandwidths which depend on the available sample. Using a specific bandwidth, we can reach a faster rate of convergence of the bias term of our estimators. Furthermore, we prove asymptotic properties like consistency and asymptotic normality. Afterwards we consider a more general model, which is driven by an α-stable Lévy motion. Again we make use of a Nadaraya-Watson like estimator for the unknown drift function. Under comparable assumptions we prove the corresponding asymptotic properties. A short simulation study illustrates our results.
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