Adaptive wavelet estimation of the diffusion coefficient under additive error measurements

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

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 1186-1216
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Hoffmann, M., Munk, A., and Schmidt-Hieber, J.. "Adaptive wavelet estimation of the diffusion coefficient under additive error measurements." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 1186-1216. <http://eudml.org/doc/272054>.

@article{Hoffmann2012,
abstract = {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.},
author = {Hoffmann, M., Munk, A., Schmidt-Hieber, J.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {adaptive estimation; Besov spaces; diffusion processes; nonparametric regression; wavelet estimation},
language = {eng},
number = {4},
pages = {1186-1216},
publisher = {Gauthier-Villars},
title = {Adaptive wavelet estimation of the diffusion coefficient under additive error measurements},
url = {http://eudml.org/doc/272054},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Hoffmann, M.
AU - Munk, A.
AU - Schmidt-Hieber, J.
TI - Adaptive wavelet estimation of the diffusion coefficient under additive error measurements
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 1186
EP - 1216
AB - 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.
LA - eng
KW - adaptive estimation; Besov spaces; diffusion processes; nonparametric regression; wavelet estimation
UR - http://eudml.org/doc/272054
ER -

References

top
  1. [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. Zbl1151.62365
  2. [2] Y. Ait-Sahalia, P. A. Mykland and L. Zhang. Ultra high frequency volatility estimation with dependent microstructure noise. J. Econometrics160 (2011) 160–175. MR2745875
  3. [3] F. M. Bandi and J. R. Russell. Separating microstructure noise from volatility. J. Financ. Econom.79 (2006) 655–692. 
  4. [4] F. M. Bandi and J. R. Russell. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. J. Econometrics160 (2011) 145–159. MR2745874
  5. [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. Econometrica76 (2008) 1481–1536. Zbl1153.91416MR2468558
  6. [6] B. Bercu and A. Touati. Exponential inequalities for self-normalized martingales with application. Ann. Appl. Probab.18 (2008) 1848–1869. Zbl1152.60309MR2462551
  7. [7] T. Cai, A. Munk and J. Schmidt-Hieber. Sharp minimax estimation of the variance of Brownian motion corrupted with Gaussian noise. Statist. Sinica20 (2010) 1011–1024. Zbl05769953MR2729850
  8. [8] Z. Ciesielski, G. Kerkyacharian and B. Roynette. Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math.107 (1993) 171–204. Zbl0809.60004MR1244574
  9. [9] A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam, 2003. Zbl1038.65151MR1990555
  10. [10] A. Cohen, I. Daubechies and P. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal.1 (1993) 54–81. Zbl0795.42018MR1256527
  11. [11] F. X. Diebold and G. H. Strasser. On the correlation structure of microstructure noise in theory and practice. Working paper, 2008. 
  12. [12] S. Donnet and A. Samson. Estimation of parameters in incomplete data models defined by dynamical systems. J. Statist. Plann. Inference137 (2007) 2815–2831. Zbl1331.62099MR2323793
  13. [13] S. Donnet and A. Samson. Parametric inference for mixed models defined by stochastic differential equations. ESAIM Probab. Stat.12 (2008) 196–218. Zbl1182.62164MR2374638
  14. [14] D. Donoho and I. M. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika81 (1994) 425–455. Zbl0815.62019MR1311089
  15. [15] D. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard. Wavelet shrinkage: Asymptopia? J. Roy Statist. Soc. Ser. B57 (1995) 301–369. Zbl0827.62035MR1323344
  16. [16] J. Fan. A selective overview of nonparametric methods in financial econometrics. Statist. Sci.20 (2005) 317–357. Zbl1130.62364MR2210224
  17. [17] B. Favetto and A. Samson. Parameter estimation for a bidimensional partially observed Ornstein–Uhlenbeck. Scand. J. Stat.37 (2010) 200–220. Zbl1224.62032MR2682296
  18. [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. Zbl0776.62033MR1211787
  19. [19] A. Gloter and M. Hoffmann. Estimation of the Hurst parameter from discrete noisy data. Ann. Statist.35 (2007) 1947–1974. Zbl1126.62073MR2363959
  20. [20] A. Gloter and J. Jacod. Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Stat. 5 (2001) 225–242. Zbl1008.60089MR1875672
  21. [21] A. Gloter and J. Jacod. Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab. Stat. 5 (2001) 243–260. Zbl1009.60065MR1875673
  22. [22] P. Hall and C. C. Heyde. Martingale Limit Theory and Its Applications. Academic Press, New York, 1980. Zbl0462.60045MR624435
  23. [23] W. Härdle, D. Picard, A. Tsybakov and G. Kerkyacharian. Wavelets, Approximation and Statistical Applications. Springer, Berlin, 1998. Zbl0899.62002MR1618204
  24. [24] M. Hoffmann. Minimax estimation of the diffusion coefficient through irregular sampling. Statist. Probab. Lett.32 (1997) 11–24. Zbl0872.62080MR1439493
  25. [25] M. Hoffmann. Adaptive estimation in diffusion processes. Stochastic Process. Appl.79 (1999) 135–163. Zbl1043.62528MR1670522
  26. [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. Zbl1166.62078MR2531091
  27. [27] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1998. Zbl0734.60060MR917065
  28. [28] G. Kerkyacharian and D. Picard. Thresholding algorithms, maxisets and well-concentrated bases. Test9 (2000) 283–345. Zbl1107.62323MR1821645
  29. [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. MR2798510
  30. [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. Zbl1329.62366MR2684388
  31. [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. Zbl1130.62364
  32. [32] M. Podolskij and M. Vetter. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli15 (2009) 634–658. Zbl1200.62131MR2555193
  33. [33] M. Reiß. Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist.39 (2011) 772–802. Zbl1215.62113MR2816338
  34. [34] M. Rosenbaum. Estimation of the volatility persistence in a discretely observed diffusion model. Stochastic Process. Appl.118 (2008) 1434–1462. Zbl1142.62055MR2427046
  35. [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. Zbl1274.62035
  36. [36] E. Schmisser. Non-parametric drift estimation for diffusions from noisy data. Statist. Decisions28 (2011) 119–150. Zbl1215.62087MR2810618
  37. [37] B.-D. Seo. Realized volatility and colored market microstructure noise. Manuscript, 2005. 
  38. [38] L. Zhang. Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli12 (2006) 1019–1043. Zbl1117.62119MR2274854
  39. [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. Zbl1117.62461MR2236450

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