Efficient estimation of stable Lévy process with symmetric jumps

  • Alexandre Brouste
  • Hiroki MasudaEmail author


Efficient estimation of a non-Gaussian stable Lévy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.



The authors also thank the reviewers and the associated editor for their valuable comments, which in particular led to substantial improvements of the arguments in Sects. 3.2 and  3.4. HM especially thanks Professor Jean Jacod for letting him notice the mistake in Masuda (2009), which has been fixed in the present paper.


  1. Aït-Sahalia Y, Jacod J (2008) Fisher’s information for discretely sampled Lévy processes. Econometrica 76(4):727–761MathSciNetCrossRefzbMATHGoogle Scholar
  2. Brouste A, Fukasawa M (2016) Local asymptotic normality property for fractional gaussian noise under high-frequency observations. arXiv preprint arXiv:1610.03694 (to appear in Ann Stat)
  3. Clément E, Gloter A (2015) Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes. Stoch Process Appl 125(6):2316–2352CrossRefzbMATHGoogle Scholar
  4. Cohen S, Gamboa F, Lacaux C, Loubes J-M (2013) LAN property for some fractional type Brownian motion. ALEA Lat Am J Probab Math Stat 10(1):91–106MathSciNetzbMATHGoogle Scholar
  5. DuMouchel WH (1973) On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution. Ann Stat 1:948–957MathSciNetCrossRefzbMATHGoogle Scholar
  6. Gloter A, Jacod J (2001a) Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab Stat 5:225–242MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gloter A, Jacod J (2001b) Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab Stat 5:243–260MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gobet E (2001) Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7(6):899–912MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gobet E (2002) LAN property for ergodic diffusions with discrete observations. Ann Inst H Poincaré Probab Stat 38(5):711–737MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hájek J (1972) Local asymptotic minimax and admissibility in estimation. In: Proceedings of the 6th Berkeley symposium on mathematical statistics and probability (Univ. California, Berkeley, Calif., 1970/1971), vol I, Theory of statistics. California University, California Press, Berkeley, pp 175–194Google Scholar
  11. Höpfner R, Jacod J (1994) Some remarks on the joint estimation of the index and the scale parameter for stable processes. In: Asymptotic statistics (Prague, 1993), contributions to statistics physica, Heidelberg, pp 273–284Google Scholar
  12. Ibragimov IA, Has’minskiĭ RZ (1981) Statistical estimation, volume 16 of applications of mathematics (Asymptotic theory, Translated from the Russian by Samuel Kotz). Springer, New YorkGoogle Scholar
  13. Ivanenko D, Kulik AM, Masuda H (2015) Uniform lan property of locally stable lévy process observed at high frequency. ALEA Lat Am J Probab Math Stat 12(2):835–862MathSciNetzbMATHGoogle Scholar
  14. Kawai R, Masuda H (2013) Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling. ESAIM Probab Stat 17:13–32MathSciNetCrossRefzbMATHGoogle Scholar
  15. Le Cam L (1972) Limits of experiments. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Univ. California, Berkeley, Calif., 1970/1971), vol I, Theory of statistics. University California Press, Berkeley, pp 245–261Google Scholar
  16. Masuda H (2009) Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density. J Jpn Stat Soc 39(1):49–75CrossRefGoogle Scholar
  17. Masuda H (2010) On statistical aspects in calibrating a geometric skewed stable asset price model. In: Recent advances in financial engineering 2009: proceedings of the KIER-TMU international workshop on financial engineering 2009, pp 181–202Google Scholar
  18. Masuda H (2015) Parametric estimation of Lévy processes. In: Barndorff-Nielsen OE, Bertoin J, Jacod J, Klüppelberg C (eds) Lévy matters. IV, volume 2128 of lecture notes in mathematics. Springer, Cham, pp 179–286Google Scholar
  19. Matsui M, Takemura A (2006) Some improvements in numerical evaluation of symmetric stable density and its derivatives. Commun Stat Theory Methods 35(1–3):149–172MathSciNetCrossRefzbMATHGoogle Scholar
  20. Reiß M (2011) Asymptotic equivalence for inference on the volatility from noisy observations. Ann Stat 39(2):772–802MathSciNetCrossRefzbMATHGoogle Scholar
  21. Robust I (2017) Analysis. User manual for stable 5.3 R version. Accessed 1 Nov 2017
  22. Roussas GG (1972) Contiguity of probability measures: some applications in statistics. Cambridge tracts in mathematics and mathematical physics, No. 63. Cambridge University Press, LondonCrossRefzbMATHGoogle Scholar
  23. Sato K (1999) Lévy processes and infinitely divisible distributions, volume 68 of Cambridge studies in advanced mathematics (Translated from the 1990 Japanese original, Revised by the author). Cambridge University Press, CambridgeGoogle Scholar
  24. Sweeting TJ (1980) Uniform asymptotic normality of the maximum likelihood estimator. Ann Stat 8(6):1375–1381 (Corrections: (1982) Annals of Statistics 10, 320)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Todorov V (2013) Power variation from second order differences for pure jump semimartingales. Stoch Process Appl 123(7):2829–2850MathSciNetCrossRefzbMATHGoogle Scholar
  26. Zolotarev VM (1986) One-dimensional stable distributions, volume 65 of translations of mathematical monographs (Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver). American Mathematical Society, Providence, RIGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Manceau de MathématiquesLe Mans Université Avenue Olivier MessiaenLe Mans Cedex 9France
  2. 2.Faculty of MathematicsKyushu UniversityFukuokaJapan

Personalised recommendations