Adaptive wavelet estimation of the diffusion coefficient under additive error measurements

Hoffmann, M.; Munk, A.; Schmidt-Hieber, J.

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1186-1216.

On étudie l'estimation non-paramétrique du coefficient de diffusion à partir d'observations discrètes, lorsque les observations sont bruitées par un bruit additionnel. De tels problèmes se sont développés au cours des dix dernières années dans plusieurs champs d'application, en particuler pour la modélisation des données haute fréquence en finance, cependant plutôt d'un point de vue paramétrique ou semi-paramétrique. Ce travail concerne l'estimation de la trajectoire (éventuellement stochastique) du coefficient de diffusion dans un cadre relativement général. En développant des techniques de pré-moyennage combinées avec du seuillage des coefficients d'ondelettes, nous contruisons des estimateurs adaptatifs qui atteignent une vitesse quasi-optimale parmi une vaste échelle de contraintes de régularité de type Besov. Puisque le coefficient de diffusion est souvent intrinsèquement aléatoire, nous proposons un nouveau critère pour qualifier la qualité d'estimation ; nous retrouvons la théorie minimax usuelle lorsque cette approche est restreinte à un coefficient de diffusion déterministe. En particulier, on exploite les résultats récents de Reiß (Ann. Statist. 39 (2011) 772-802) de l'équivalence asymptotique entre une diffusion gaussienne avec un bruit additif et le bruit blanc gaussien.

We study nonparametric estimation of the diffusion coefficient from discrete data, when the observations are blurred by additional noise. Such issues have been developed over the last 10 years in several application fields and in particular in high frequency financial data modelling, however mainly from a parametric and semiparametric point of view. This paper addresses the nonparametric estimation of the path of the (possibly stochastic) diffusion coefficient in a relatively general setting. By developing pre-averaging techniques combined with wavelet thresholding, we construct adaptive estimators that achieve a nearly optimal rate within a large scale of smoothness constraints of Besov type. Since the diffusion coefficient is usually genuinely random, we propose a new criterion to assess the quality of estimation; we retrieve the usual minimax theory when this approach is restricted to a deterministic diffusion coefficient. In particular, we take advantage of recent results of Reiß (Ann. Statist. 39 (2011) 772-802) of asymptotic equivalence between a Gaussian diffusion with additive noise and Gaussian white noise model, in order to prove a sharp lower bound.

[1] Y. Ait-Sahalia, P. A. Mykland and L. Zhang. How often to sample a continuous-time process in the presence of market microstructure noise. Rev. Financ. Stud. 18 (2005) 351-416. | Zbl

[2] Y. Ait-Sahalia, P. A. Mykland and L. Zhang. Ultra high frequency volatility estimation with dependent microstructure noise. J. Econometrics 160 (2011) 160-175. | MR

[3] F. M. Bandi and J. R. Russell. Separating microstructure noise from volatility. J. Financ. Econom. 79 (2006) 655-692.

[4] F. M. Bandi and J. R. Russell. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. J. Econometrics 160 (2011) 145-159. | MR

[5] O. Barndorff-Nielsen, P. Hansen, A. Lunde and N. Shephard. Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76 (2008) 1481-1536. | MR | Zbl

[6] B. Bercu and A. Touati. Exponential inequalities for self-normalized martingales with application. Ann. Appl. Probab. 18 (2008) 1848-1869. | MR | Zbl

[7] T. Cai, A. Munk and J. Schmidt-Hieber. Sharp minimax estimation of the variance of Brownian motion corrupted with Gaussian noise. Statist. Sinica 20 (2010) 1011-1024. | MR

[8] Z. Ciesielski, G. Kerkyacharian and B. Roynette. Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math. 107 (1993) 171-204. | MR | Zbl

[9] A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam, 2003. | MR | Zbl

[10] A. Cohen, I. Daubechies and P. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 (1993) 54-81. | MR | Zbl

[11] F. X. Diebold and G. H. Strasser. On the correlation structure of microstructure noise in theory and practice. Working paper, 2008.

[12] S. Donnet and A. Samson. Estimation of parameters in incomplete data models defined by dynamical systems. J. Statist. Plann. Inference 137 (2007) 2815-2831. | MR

[13] S. Donnet and A. Samson. Parametric inference for mixed models defined by stochastic differential equations. ESAIM Probab. Stat. 12 (2008) 196-218. | Numdam | MR | Zbl

[14] D. Donoho and I. M. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 (1994) 425-455. | MR | Zbl

[15] D. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard. Wavelet shrinkage: Asymptopia? J. Roy Statist. Soc. Ser. B 57 (1995) 301-369. | MR | Zbl

[16] J. Fan. A selective overview of nonparametric methods in financial econometrics. Statist. Sci. 20 (2005) 317-357. | MR | Zbl

[17] B. Favetto and A. Samson. Parameter estimation for a bidimensional partially observed Ornstein-Uhlenbeck. Scand. J. Stat. 37 (2010) 200-220. | MR | Zbl

[18] V. Genon-Catalot, C. Laredo and D. Picard. Non-parametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Stat. 19 (1992) 317-335. | MR | Zbl

[19] A. Gloter and M. Hoffmann. Estimation of the Hurst parameter from discrete noisy data. Ann. Statist. 35 (2007) 1947-1974. | MR | Zbl

[20] A. Gloter and J. Jacod. Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Stat. 5 (2001) 225-242. | Numdam | MR | Zbl

[21] A. Gloter and J. Jacod. Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab. Stat. 5 (2001) 243-260. | Numdam | MR | Zbl

[22] P. Hall and C. C. Heyde. Martingale Limit Theory and Its Applications. Academic Press, New York, 1980. | MR | Zbl

[23] W. Härdle, D. Picard, A. Tsybakov and G. Kerkyacharian. Wavelets, Approximation and Statistical Applications. Springer, Berlin, 1998. | MR | Zbl

[24] M. Hoffmann. Minimax estimation of the diffusion coefficient through irregular sampling. Statist. Probab. Lett. 32 (1997) 11-24. | MR | Zbl

[25] M. Hoffmann. Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 (1999) 135-163. | MR | Zbl

[26] J. Jacod, Y. Li, P. A. Mykland, M. Podolskij and M. Vetter. Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process. Appl. 119 (2009) 2249-2276. | MR | Zbl

[27] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1998. | MR | Zbl

[28] G. Kerkyacharian and D. Picard. Thresholding algorithms, maxisets and well-concentrated bases. Test 9 (2000) 283-345. | MR | Zbl

[29] A. Munk and J. Schmidt-Hieber. Lower bounds for volatility estimation in microstructure noise models. In Borrowing Strength: Theory Powering Applications - A Festschrift for Lawrence D. Brown 43-55. Inst. Math. Stat. Collect. 6. Institute of Mathematical Statistics, Beachwood, OH, 2010. | MR

[30] A. Munk and J. Schmidt-Hieber. Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise. Electron. J. Stat. 4 (2010) 781-821. | MR

[31] P. A. Mykland and L. Zhang. Discussion of “A selective overview of nonparametric methods in financial econometrics” by Jianqing Fan. Statist. Sci. 20 (2005) 347-350. | Zbl

[32] M. Podolskij and M. Vetter. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15 (2009) 634-658. | MR | Zbl

[33] M. Reiß. Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 (2011) 772-802. | MR | Zbl

[34] M. Rosenbaum. Estimation of the volatility persistence in a discretely observed diffusion model. Stochastic Process. Appl. 118 (2008) 1434-1462. | MR | Zbl

[35] J. Schmidt-Hieber. Nonparametric methods in spot volatility estimation. Ph.D. thesis, 2011. Available at http://webdoc.sub.gwdg.de/diss/2011/schmidt_hieber. | Zbl

[36] E. Schmisser. Non-parametric drift estimation for diffusions from noisy data. Statist. Decisions 28 (2011) 119-150. | MR | Zbl

[37] B.-D. Seo. Realized volatility and colored market microstructure noise. Manuscript, 2005.

[38] L. Zhang. Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12 (2006) 1019-1043. | MR | Zbl

[39] L. Zhang, P. Mykland and Y. Ait-Sahalia. A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 (2005) 1394-1411. | MR | Zbl