Geodesic
Improved nonparametric estimation of the drift in diffusion processes
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 364-372 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we have considered the robust adaptive nonparametric estimation problem for the drift coefficient in diffusion processes. It has been shown that the initial estimation problem can be reduced to the estimation problem in a discrete time nonparametric heteroscedastic regression model by using the sequential approach. We have developed a new sharp model selection method for estimating the unknown drift function using the improved estimation approach. An adaptive model selection procedure based on the improved weighted least square estimates has been proposed. It has been established that such estimate outperforms in non-asymptotic mean square accuracy the procedure based on the classical weighted least square estimates. Sharp oracle inequalities for the robust risk have been obtained.
Keywords: improved estimation, mean-square accuracy, model selection, sharp oracle inequality.
Mots-clés : stochastic diffusion process 
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     title = {Improved nonparametric estimation of the drift in diffusion processes},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {364--372},
     year = {2018},
     volume = {160},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a17/}
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E. A. Pchelintsev; S. S. Perelevskiy; I. A. Makarova. Improved nonparametric estimation of the drift in diffusion processes. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 364-372. http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a17/

[1] Karatzas I., Shreve S. E., Methods of Mathematical Finance, Springer, New York, 1998, xv+415 pp. | DOI | MR | Zbl

[2] Kutoyants Yu. A., Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004, xiv+482 pp. | DOI | MR | Zbl

[3] Galtchouk L. I., Konev V. V., “On sequential estimation of parameters in continuous-time stochastic regression”, Statistics and Control of Stochastic Processes, World Sci. Publ., River Edge, 1997, 123–138 | MR | Zbl

[4] Galtchouk L. I., Konev V. V., “On sequential estimation of parameters in semimartingale regression models with continuous time parameter”, Ann. Stat., 29 (2001), 1508–2035 | DOI | MR

[5] Konev V. V., Pergamenshchikov S. M., “On truncated sequential estimation of the parameters of diffusion processes”, Methods of Economical Analysis, Tsentr. Ekon.-Mat. Inst. Ross. Akad. Nauk, M., 1992, 3–31 (In Russian) | MR

[6] Galtchouk L. I., Pergamenshchikov S. M., “Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes”, Stat. Inference Stoch. Processes, 9:1 (2006), 1–16 | DOI | MR | Zbl

[7] Pchelintsev E., “Improved estimation in a non-Gaussian parametric regression”, Stat. Inference Stoch. Processes, 16:1 (2013), 15–28 | DOI | MR | Zbl

[8] Konev V. V., Pergamenshchikov S. M., Pchelintsev E. A., “Estimation of a regression with the pulse type noise from discrete data”, Theory Probab. Its Appl., 58:3 (2014), 442–457 | DOI | MR | Zbl

[9] Pchelintsev E., Pchelintsev V., Pergamenshchikov S., Improved robust model selection methods for the Lévy nonparametric regression in continuous time, 2017, 32 pp., arXiv: 1710.03111

[10] Galtchouk L. I., Pergamenshchikov S. M., Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 1. Sharp non-asymptotic oracle inequalities, , 34 pp. hal-00177875

[11] Ibragimov I. A., Has'minskii R. Z., Statistical Estimation: Asymptotic Theory, Springer, New York, 1981, vii+403 pp. | DOI | MR | Zbl