Two-step estimation of ergodic Lévy driven SDE



We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient \(a(x,\alpha )\) and scale coefficient \(c(x,\gamma )\) involving unknown parameters \(\alpha \) and \(\gamma \). We suppose that the Lévy measure \(\nu _{0}\), has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of \(\alpha \), \(\gamma \) and a class of functional parameter \(\int \varphi (z)\nu _0(dz)\), which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of \((\alpha ,\gamma )\); and then, for estimating \(\int \varphi (z)\nu _0(dz)\) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.


Asymptotic normality Ergodicity Functional parameter estimation Gaussian quasi-likelihood estimation High-frequency sampling Lévy driven stochastic differential equation 



We are grateful to the referees for careful reading and constructive comments, which led to substantial improvements of the earlier version of this paper. This work was partly supported by JSPS KAKENHI Grant Numbers 26400204 (H. Masuda) and JST, CREST.


  1. Applebaum D (2009) Lévy processes and stochastic calculus. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  2. Brouste A, Fukasawa M, Hino H, Iacus S, Kamatani K, Koike Y, Masuda H, Nomura R, Ogihara T, Shimuzu Y, Uchida M, Yoshida N (2014) The YUIMA project: a computational framework for simulation and inference of stochastic differential equations. J Stat Softw 57:1–51Google Scholar
  3. Dvoretzky A (1972) Asymptotic normality for sums of dependent random variables. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (University of California, Berkeley, California, 1970/1971), Probability theory, vol II. University of California Press, Berkeley, pp 513–535Google Scholar
  4. Feuerverger A (1990) An efficiency result for the empirical characteristic function in stationary time-series models. Can J Stat 18:155–161MathSciNetCrossRefMATHGoogle Scholar
  5. Figueroa-López JE (2008) Small-time moment asymptotics for Lévy processes. Stat Prob Lett 78:3355–3365CrossRefMATHGoogle Scholar
  6. Figueroa-López JE (2009) Nonparametric estimation for Lévy models based on discrete-sampling. Lecture notes-monograph series, pp 117–146Google Scholar
  7. Genon-Catalo V, Jacod J (1993) On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 29:119–151MathSciNetMATHGoogle Scholar
  8. Gobet E (2002) LAN property for ergodic diffusions with discrete observations. Annales de l’Institut Henri Poincare (B) Probability and Statistics 38:711–737MathSciNetCrossRefMATHGoogle Scholar
  9. Jacod J (2007) Asymptotic properties of power variations of Lévy processes. ESAIM Prob Stat 11:173–196CrossRefMATHGoogle Scholar
  10. Kessler M (1997) Estimation of an ergodic diffusion from discrete observations. Scand J Stat 24:211–229MathSciNetCrossRefMATHGoogle Scholar
  11. Kunita H (1997) Stochastic flows and stochastic differential equations, 24th edn. Cambridge University Press, CambridgeMATHGoogle Scholar
  12. Kutoyants YA (2004) Statistical inference for ergodic diffusion processes. Springer, New YorkCrossRefMATHGoogle Scholar
  13. Liptser RS, Shiryaev AN (2001) Statistics of Random Processes II: II. Applications, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  14. Luschgy H, Pagès G (2008) Moment estimates for Lévy processes. Electron Commun Prob 13:422–434CrossRefMATHGoogle Scholar
  15. Masuda H (2013) Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann Stat 41:1593–1641CrossRefMATHGoogle Scholar
  16. Prasaka Rao B (1999) Statistical inference for diffusion type processes. Arnold, LondonGoogle Scholar
  17. R Development Core Team: R (2010) A language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  18. Sato K-I (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, CambridgeMATHGoogle Scholar
  19. Shimizu Y (2009) Functional estimation for Levy measures of semimartingales with Poissonian jumps. J Multivar Anal 100:1073–1092CrossRefMATHGoogle Scholar
  20. van der Vaart AW (2000) Asymptotic statistics. Cambridge University Press, CambridgeMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Graduate School of MathematicsKyushu UniversityFukuokaJapan

Personalised recommendations