×

Adaptive estimation in diffusion processes. (English) Zbl 1043.62528

Summary: We study the nonparametric estimation of the coefficients of a 1-dimensional diffusion process from discrete observations. Different asymptotic frameworks are considered. Minimax rates of convergence are studied over a wide range of Besov smoothness classes. We construct estimators based on wavelet thresholding which are adaptive (with respect to an unknown degree of smoothness). The results are comparable with simpler models such as density estimation or nonparametric regression.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Barlow, M.T.; Yor, M., Semi-martingale inequalities via the garsia-rodemich-rumsey lemma and application to local times, J. funct. anal., 49, 198-229, (1982) · Zbl 0505.60054
[2] Cohen, A., 1998. Multiscale methods and wavelets in numerical analysis. Handbook of Numer. Anal., to appear.
[3] Daubechies, I., 1992. Ten lectures on wavelets, ‘CBMS-NSF Series in Applied Mathematics’, vol. 61, SIAM, Philadelphia. · Zbl 0776.42018
[4] Donoho, D.L., Johnstone, I.M., 1995. Minimax estimation via wavelet shrinkage, Ann. Statist., to appear. · Zbl 0935.62041
[5] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D., 1995. Wavelet shrinkage, Asymptopia ? J. Roy. Statist. Soc. Ser. B 57, 301-369. With discussion. · Zbl 0827.62035
[6] Donoho, D.L.; Johnstone, I.M.; Kerkyacharian, G.; Picard, D., Density estimation by wavelet thresholding, Ann. statist., 24, 508-539, (1996) · Zbl 0860.62032
[7] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, J. appl. probab., 30, 790-804, (1993) · Zbl 0796.62070
[8] Genon-Catalot, V.; Laredo, C.; Picard, D., Nonparametric estimation of the diffusion coefficient by wavelet methods, Scand. J. statist., 19, 319-335, (1992) · Zbl 0776.62033
[9] Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and Its Applications. Academic Press, New York. · Zbl 0462.60045
[10] Hoffmann, M., 1996. Lp estimation of the diffusion coefficient. Bernoulli, to appear.
[11] Jacod, J., 1997. Nonparametric kernel estimation of the diffusion coefficient of a diffusion. Preprint 405, Laboratoire de Probabilités, Université Paris 6.
[12] Korostelev, A.P., Tsybakov, A.D., 1993. Minimax Theory of Image Reconstruction. Lecture Notes in Statistics, vol. 82. Springer, Berlin. · Zbl 0833.62039
[13] Kerkyacharian, G.; Picard, D., Density estimation by kernel and wavelets methodes:optimality of Besov spaces, Statist. probab. lett., 18, 327-336, (1993) · Zbl 0793.62019
[14] Lipster, R.S., Shiryayev, A.N., 1977. Statistics of random processes I: General Theory. Springer, Berlin.
[15] Meyer, Y., Ondelettes et opérateurs I, (1990), Hermann Paris
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.