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Oracle inequalities for the stochastic differential equations

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Abstract

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi-martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein–Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.

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References

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723

    Article  MathSciNet  MATH  Google Scholar 

  • Barbu VS, Limnios N (2008) Semi-Markov chains and hidden semi-Markov models toward applications: their use in reliability and DNA analysis. Lecture notes in statistics, vol 191. Springer, Berlin

    MATH  Google Scholar 

  • Barbu V, Beltaif S, Pergamenshchikov SM (2017a) Robust adaptive efficient estimation for semi-Markov nonparametric regression models. Preprint. https://arxiv.org/pdf/1604.04516.pdf (submitted in Statistical inference for stochastic processes)

  • Barbu V, Beltaif S, Pergamenshchikov SM (2017b) Robust adaptive efficient estimation for a semi-Markov continuous time regression from discrete data. Preprint. http://arxiv.org/abs/1710.10653 (submitted in Annales de l’Institut Henri Poincaré)

  • Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial mathematics. J R Stat Soc B63:167–241

    Article  MATH  Google Scholar 

  • Barron A, Birgé L, Massart P (1999) Risk bounds for model selection via penalization. Probab Theory Relat Fields 113:301–415

    Article  MathSciNet  MATH  Google Scholar 

  • Bertoin J (1996) Lévy processes. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Hall, London

    MATH  Google Scholar 

  • Delong L, Klüppelberg C (2008) Optimal investment and consumption in a Black–Scholes market with Lévy driven stochastic coefficients. Ann Appl Probab 18(3):879–908

    Article  MathSciNet  MATH  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer series in statistics. Springer, New York

    MATH  Google Scholar 

  • Fourdrinier D, Pergamenshchikov S (2007) Improved selection model method for the regression with dependent noise. Ann Inst Stat Math 59(3):435–464

    Article  MATH  Google Scholar 

  • Fourdrinier D, Strawderman WE (1996) A paradox concerning shrinkage estimators: should a known scale parameter be replaced by an estimated value in the shrinkage factor? J Multivar Anal 59(2):109–140

    Article  MathSciNet  MATH  Google Scholar 

  • Galtchouk LI, Pergamenshchikov SM (2006) Asymptotically efficient estimates for non parametric regression models. Stat Probab Lett 76(8):852–860

    Article  MATH  Google Scholar 

  • Galtchouk LI, Pergamenshchikov SM (2009) Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models. J Nonparametric Stat 21(1):1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Galtchouk LI, Pergamenshchikov SM (2009) Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression. J Korean Stat Soc 38(4):305–322

    Article  MathSciNet  MATH  Google Scholar 

  • Goldie CM (1991) Implicit renewal theory and tails of solutions of random equations. Ann Appl Probab 1(1):126–166

    Article  MathSciNet  MATH  Google Scholar 

  • Höpfner R, Kutoyants YA (2009) On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion. Stat Decis 27(4):309–326

    MathSciNet  MATH  Google Scholar 

  • Höpfner R, Kutoyants YA (2010) Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Stat Infer Stoch Process 13(3):193–230

    Article  MathSciNet  MATH  Google Scholar 

  • Ibragimov IA, Khasminskii RZ (1981) Statistical estimation: asymptotic theory. Springer, New York

    Book  Google Scholar 

  • Jacod J, Shiryaev AN (2002) Limit theorems for stochastic processes, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • James W, Stein C (1961) Estimation with quadratic loss. In: Proceedings of the fourth berkeley symposium mathematics, statistics and probability, University of California Press, Berkeley, vol 1, pp 361–380

  • Kassam SA (1988) Signal detection in non-Gaussian noise. Springer, New York

    Book  Google Scholar 

  • Kneip A (1994) Ordered linear smoothers. Ann Stat 22:835–866

    Article  MathSciNet  MATH  Google Scholar 

  • Konev VV, Pergamenshchikov SM (2003) Sequential estimation of the parameters in a trigonometric regression model with the Gaussian coloured noise. Stat Inference Stoch Process 6:215–235

    Article  MathSciNet  MATH  Google Scholar 

  • Konev VV, Pergamenshchikov SM (2009) Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities. J Math Mech Tomsk State Univ 3:23–41

    Google Scholar 

  • Konev VV, Pergamenshchikov SM (2009) Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency. J Math Mech Tomsk State Univ 4:31–45

    Google Scholar 

  • Konev VV, Pergamenshchikov SM (2010) General model selection estimation of a periodic regression with a Gaussian noise. Ann Inst Stat Math 62:1083–1111

    Article  MathSciNet  MATH  Google Scholar 

  • Konev VV, Pergamenshchikov SM (2012) Efficient robust nonparametric estimation in a semimartingale regression model. Ann Inst Henri Poincaré Probab Stat 48(4):1217–1244

    Article  MathSciNet  MATH  Google Scholar 

  • Konev VV, Pergamenshchikov SM (2015) Robust model selection for a semimartingale continuous time regression from discrete data. Stoch Process Appl 125:294–326

    Article  MathSciNet  MATH  Google Scholar 

  • Konev VV, Pergamenshchikov SM, Pchelintsev E (2014) Estimation of a regression with the pulse type noise from discrete data. Theory Probab Appl 58(3):442–457

    Article  MathSciNet  MATH  Google Scholar 

  • Kutoyants YA (1977) Estimation of the signal parameter in a Gaussian Noise. Probl Inf Transm 13(4):29–36

    MathSciNet  MATH  Google Scholar 

  • Kutoyants YA (1984) Parameter estimation for stochastic processes. Heldeman, Berlin

    MATH  Google Scholar 

  • Limnios N, Oprisan G (2001) Semi-Markov processes and reliability. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  • Mallows C (1973) Some comments on \(C_{p}\). Technometrics 15:661–675

    MATH  Google Scholar 

  • Mikosch T (2004) Non-life insurance mathematics. An introduction with stochastic processes. Springer, Berlin

    MATH  Google Scholar 

  • Nussbaum M (1985) Spline smoothing in regression models and asymptotic efficiency in \({\bf L}_2\). Ann Stat 13:984–997

    Article  MATH  Google Scholar 

  • Pchelintsev EA, Pchelintsev VA, Pergamenshchikov SM (2017) Improved robust model selection methods for the Levy nonparametric regression in continuous time. Preprint. http://arxiv.org/abs/1710.03111 (submitted in Stochastic Processes and their Applications)

  • Pchelintsev E (2013) Improved estimation in a non-Gaussian parametric regression. Stat Inference Stoch Process 16(1):15–28

    Article  MathSciNet  MATH  Google Scholar 

  • Pinsker MS (1981) Optimal filtration of square integrable signals in Gaussian white noise. Probl Transm Inf 17:120–133

    Google Scholar 

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Acknowledgements

The first author is partially supported by the the RSF Grant 17-11-01049. The last author is partially supported by the Russian Federal Professor program (Project No. 1.472.2016/1.4, Ministry of Education and Science) and by the project XterM - Feder, University of Rouen, France. Moreover, the authors are grateful to the anonymous referee for very helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Information Technologies and Business Analytics, Tomsk State University, Lenin str. 36, Tomsk, Russia, 634050

    E. A. Pchelintsev

  2. Laboratory of Mathematics LMRS, University of Rouen, Rouen, France

    S. M. Pergamenshchikov

  3. International Laboratory SSP & QF, Tomsk State University, Lenin str. 36, Tomsk, Russia, 634050

    S. M. Pergamenshchikov

  4. National Research University “MPEI”, 14 Krasnokazarmennaya, Moscow, Russia, 111250

    S. M. Pergamenshchikov

Authors
  1. E. A. Pchelintsev
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  2. S. M. Pergamenshchikov
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Correspondence to E. A. Pchelintsev.

Additional information

This work was done under the Ministry of Education and Science of the Russian Federation in the framework of the research Project No. 2.3208.2017/4.6, by RFBR Grant 16-01-00121 A and by the partial financial support of the RSF Grant Number 14-49-00079 (National Research University “MPEI” 14 Krasnokazarmennaya, 111250 Moscow, Russia).

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Pchelintsev, E.A., Pergamenshchikov, S.M. Oracle inequalities for the stochastic differential equations. Stat Inference Stoch Process 21, 469–483 (2018). https://doi.org/10.1007/s11203-018-9180-1

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