Improved nonparametric estimation of the drift in diffusion processes

E.A. Pchelintsev, S.S. Perelevskiy, I.A. Makarova

National Research Tomsk State University, Tomsk, 634050 Russia

Full text PDF

For citation: Pchelintsev E.A., Perelevskiy S.S., Makarova I.A. Improved nonparametric estimation of the drift in diffusion processes. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 364–372.

Для цитирования: Pchelintsev E.A., Perelevskiy S.S., Makarova I.A. Improved nonparametric estimation of the drift in diffusion processes // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 364–372.

Abstract

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, stochastic diffusion process, mean-square accuracy, model selection, sharp oracle inequality

Acknowledgements. This work was supported by the Russian Science Foundation (results of Section 2, project no. 17-11-01049) and by the Ministry of Education and Science of the Russian Federation (results of Section 3, project no. 2.3208.2017/4.6).

References

1. Karatzas I., Shreve S.E. Methods of Mathematical Finance. New York, Springer, 1998. xv, 415 p. doi: 10.1007/978-1-4939-6845-9.

2. Kutoyants Yu.A. Statistical Inference for Ergodic Diffusion Processes. London, Springer, 2004. xiv, 482 p. doi: 10.1007/978-1-4471-3866-2.

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

4. Galtchouk L.I., Konev V.V. On sequential estimation of parameters in semimartingale regression models with continuous time parameter. Ann. Stat., 2001, vol. 29, pp. 1508–2035.

5. Konev V.V., Pergamenshchikov S.M. On truncated sequential estimation of the parameters of diffusion processes. In: Metody ekonomicheskogo analiza [Methods of Economical Analysis]. Moscow, Tsentr. Ekon.-Mat. Inst. Ross. Akad. Nauk, 1992, pp. 3–31. (In Russian)

6. Galtchouk L.I., Pergamenshchikov S.M. Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes. Stat. Inference Stoch. Processes, 2006, vol. 9, no. 1, pp. 1–16. doi: 10.1007/s11203-005-3248-4.

7. Pchelintsev E. Improved estimation in a non-Gaussian parametric regression. Stat. Inference Stoch. Processes, 2013, vol. 16, no. 1, pp. 15–28. doi: 10.1007/s11203-013-9075-0.

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., 2014, vol. 58, no. 3, pp. 442–457. doi: 10.1137/S0040585X9798662X.

9. Pchelintsev E., Pchelintsev V., Pergamenshchikov S. Improved robust model selection methods for the Lvy nonparametric regression in continuous time. arXiv:1710.03111 , 2017, pp. 1–32. Available at: https://arxiv.org/submit/2029866.

10. Galtchouk L.I., Pergamenshchikov S.M. Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 1. Sharp non-asymptotic oracle inequalities. HAL Id hal-00177875, pp. 1–34. Available at: https://hal.archives-ouvertes.fr/hal-00177875.

11. Ibragimov I.A., Has'minskii R.Z. Statistical Estimation: Asymptotic Theory. New York, Springer, 1981. vii, 403 p. doi: 10.1007/978-1-4899-0027-2.

Recieved

October 24, 2017

Pchelintsev Evgeniy Anatolevich, Candidate of Physics and Mathematics, Associate Professor of the Department of Mathematical Analysis and Theory of Functions

National Research Tomsk State University

pr. Lenin, 36, Tomsk, 634050 Russia

E-mail:  evgen-pch@yandex.ru

Perelevsky Svyatoslav Sergeevich, PhD Student of the Department of Mathematical Analysis and Theory of Functions

National Research Tomsk State University

pr. Lenin, 36, Tomsk, 634050 Russia

E-mail:  slavaperelevskiy@mail.ru

Makarova Irina Alekseevna, Student of the Department of Mathematical Analysis and Theory of Functions

National Research Tomsk State University

pr. Lenin, 36, Tomsk, 634050 Russia

E-mail:  starirish@bk.ru

Контент доступен под лицензией Creative Commons Attribution 4.0 License.