Penalized nonparametric mean square estimation of the coefficients of diffusion processes

Bernoulli

Volumn 13, Issue 2, 2007, Pages 514-543

  Comte, Fabienne ^a   Genon Catalot, Valentine ^a   Rozenholc, Yves ^a  

^a UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

Adaptive estimation; Diffusion processes; Discrete time observations; Drift and diffusion coefficients; Mean square estimator; Model selection; Penalized contrast; Retrospective simulation

Indexed keywords

EID: 47249110597     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/07-BEJ5173     Document Type: Article

References (32)

1. Abramowitz, M. and Stegun, I.A. (eds) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Wiley. MR0167642
2. Banon, G. (1978). Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 380-395.
3. Baraud, Y. (2002). Model selection for regression on a random design. ESAIM Probab. Statist. 6 127-146.
4. Baraud, Y., Comte, F. and Viennet, G. (2001). Adaptive estimation in an autoregression and a geometrical β-mixing regression framework. Ann. Statist. 29 839-875.
5. Baraud, Y., Comte, F and Viennet, G. (2001). Model selection for (auto)-regression with dependent data. ESAIM Probab. Statist. 5 33-49.
6. Barlow, M.T. and Yor, M. (1982). Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma and applications to local times. J. Funct. Anal. 49 198-229.
7. Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5 151-157.
8. Barron, A.R., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301-413.
9. Beskos, A. and Roberts, G.O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422-2444.
10. Beskos, A., Papaspiliopoulos, O. and Roberts, G.O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077-1098.
11. Bibby, B.M. and Sørensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 17-39.
12. Bibby, B.M., Jacobsen, M. and Sørensen, M. (2002). Estimating functions for discretely sampled diffusion-type models. In Handbook of Financial Econometrics. Amsterdam: North-Holland.
13. Birgé, L. and Massart, P. (1997). From model selection to adaptive estimation. In D. Pllard, E. Torgessen and G.L. Yang (eds), Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, pp. 55-87. New York: Springer-Verlag.
14. Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 329-375.
15. Comte, F. (2001). Adaptive estimation of the spectrum of a stationary Gaussian sequence. Bernoulli 7 267-298.
16. Comte, F. and Rozenholc, Y. (2002). Adaptive estimation of mean and volatility functions in (auto-)regressive models. Stochastic Process. Appl. 97 111-145.
17. Comte, F. and Rozenholc, Y. (2004). A new algorithm for fixed design regression and denoising. Ann. Inst. Statist. Math. 56 449-473.
18. Dalalyan, A. (2005). Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 2507-2528.
19. Daubechies, I. (1992). Ten lectures on wavelets. CBMS-NSF Regional Conference Series in applied Mathematics, 6I. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
20. Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
21. Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 790-804.
22. Genon-Catalot, V., Larédo, C. and Picard, D. (1992). Nonparametric estimation of the diffusion coefficient by wavelet methods. Scand. J. Statist. 19 319-335.
23. Gloter, A. (2000). Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM Probab. Statist. 4 205-227.
24. Gobet, E., Hoffmann, M. and Reiss, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 2223-2253.
25. Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 135-163.
26. Jacod, J. (2000). Non-parametric kernel estimation of the coefficient of a diffusion. Scand. J. Statist. 27 83-96.
27. Kessler, M. and Sørensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5 299-314.
28. Kutoyants, Y.A. (2004). Statistical Inference for Ergodic Diffusion Processes. London: Springer-Verlag.
29. Pardoux, E. and Veretennikov, A. Yu. (2001). On the Poisson equation and diffusion approximation. I. Ann. Probab. 29 1061-1085.
30. Prakasa Rao, B.L.S. (1999). Statistical Inference for Diffusion Type Processes. London: Edward Arnold.
31. Spokoiny, V.G. (2000). Adaptive drift estimation for nonparametric diffusion model. Ann. Statist. 28 815-836.
32. Viennet, G. (1997). Inequalities for absolutely regular sequences: application to density estimation. Probab. Theory Related Fields 107 467-492.