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On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes

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Abstract

In this paper we give easy to verify conditions for the strong consistency of the maximum likelihood estimator (MLE) in the case when data is sampled from a parametric family of selfdecomposable distributions. The difficulty arises from the fact that standard conditions for the consistency of the MLE are based on the pdf, which, for most selfdecomposable distributions, is not available in a closed form. Instead, our conditions are based on properties of the Lévy triplet (i.e. the Lévy measure, the Gaussian part, and the shift) of the distribution. Further, we extend out results to certain selfdecomposable stochastic processes, and, in particular, we give conditions in the case when the data is sampled from a Lévy or an Ornstein–Uhlenbeck process.

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References

  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions, 9th edn. Dover Publications, New York

  2. Aoyama T, Maejima M, Rosiński J (2008) A subclass of type \(G\) selfdecomposable distributions on \({\mathbb{R}}^d\). J Theor Probab 21:14–34

  3. Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B 62(2):167–241

  4. Barndorff-Nielsen OE, Shephard N (2002a) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J R Stat Soc Ser B 64(2):253–280

  5. Barndorff-Nielsen OE, Shephard N (2002b) Normal modified stable processes. Theory Probab Math Stat 65:1–20

  6. Bianchi ML, Rachev ST, Kim YS, Fabozzi FJ (2010) Tempered stable distributions and processes in finance: numerical analysis. In: Corazza M, Pizzi C (eds) Mathematical and statistical methods for actuarial sciences and finance. Springer, Dordrecht, pp 33–42

  7. Bianchi ML, Rachev ST, Fabozzi FJ (2014) Tempered stable Ornstein–Uhlenbeck processes: a practical view. Bank of Italy Temi di Discussione, Working Paper No. 912

  8. Brorsen BW, Yang SR (1990) Maximum likelihood estimates of symmetric stable distribution parameters. Commun Stat Simul Comput 19(4):1459–1464

  9. Cao L, Grabchak M (2014) Smoothly truncated Lévy flights: toward a realistic mobility model. IPCCC ’14: Proceedings of the 33rd International Performance Computing and Communications Conference, p 8

  10. Carr P, Geman H, Madan DB, Yor M (2002) The fine structure of asset returns: an empirical investigation. J Bus 75(2):305–332

  11. Cont R, Tankov P (2004) Financial modeling with jump processes. Chapman & Hall, Boca Raton

  12. DuMouchel WH (1973) On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution. Ann Stat 1(5):948–952

  13. Eberlein E (2001) Application of generalized hyperbolic Lévy motions to finance. In: Barndorff-Nielsen OE, Mikosch T, Resnick SI (eds) Lévy processes: theory and applications. Birkäuser, Boston, pp 319–336

  14. Genon-Catalot V, Jeantheau T, Larédo C (2000) Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6(6):1051–1079

  15. Grabchak M (2012) On a new class of tempered stable distributions: moments and regular variation. J Appl Probab 49(4):1015–1035

  16. Grabchak M (2014) Does value-at-risk encourage diversification when losses follow tempered stable or more general Lévy processes? Ann Financ 10(4):553–568

  17. Grabchak M, Molchanov S (2014) Limit theorems and phase transitions for two models of summation of i.i.d. random variables with a parameter. Teoriya Veroyatnostei i ee Primeneniya 59(2):340–364

  18. Grabchak M, Samorodnitsky G (2010) Do financial returns have finite or infinite variance? A paradox and an explanation. Quant Financ 10(8):883–893

  19. Hougaard P (1986) Survival models for heterogeneous populations derived from stable distributions. Biometrika 73(2):387–396

  20. Kallenberg O (2002) Foundations of modern probability, 2nd edn. Springer, New York

  21. Kawai R, Masuda H (2011) Exact discrete sampling of finite variation tempered stable Ornstein–Uhlenbeck processes. Monte Carlo Methods Appl 17(3):279–300

  22. Kawai R, Masuda H (2012) Infinite variation tempered stable Ornstein–Uhlenbeck processes and discrete observations. Commun Stat Simul Comput 41(1):125–130

  23. Koponen I (1995) Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys Rev E 52(1):1197–1199

  24. Küchler U, Tappe S (2008a) Bilateral gamma distributions and processes in financial mathematics. Stoch Process Appl 118(2):261–283

  25. Küchler U, Tappe S (2008b) On the shapes of bilateral Gamma densities. Stat Probab Lett 78(15):2478–2484

  26. Küchler U, Tappe S (2013) Tempered stable distributions and processes. Stoch Process Appl 123(12):4256–4293

  27. Mantegna RN, Stanley HE (1994) Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys Rev Lett 73(22):2946–2949

  28. Masuda H (2007) Ergodicity and exponential \(\beta \)-mixing for multidimensional diffusions with jumps. Stoch Process Appl 117(1):35–56

  29. McCulloch JH (1998) Linear regression with stable disturbances. In: Adler R, Feldman R, Taqqu M (eds) A practical guide to heavy tails. Birkäuser, Boston, pp 359–376

  30. Neumann MH, Reiß M (2009) Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15(1):223–248

  31. Nolan JP (2001) Maximum likelihood estimation and diagnostics for stable distributions. In: Barndorff-Nielsen OE, Mikosch T, Resnick SI (eds) Lévy processes: theory and applications. Birkäuser, Boston, pp 379–400

  32. Palmer KJ, Ridout MS, Morgan BJT (2008) Modelling cell generation times by using the tempered stable distribution. J R Stat Soc Ser C (Appl Stat) 57(4):379–397

  33. Rachev ST, Kim YS, Bianchi ML, Fabozzi FJ (2011) Financial models with levy processes and volatility clustering. John Wiley & Sons Ltd., Chichester

  34. Rachev ST, Mittnik S (2000) Stable paretian models in finance. John Wiley & Sons Ltd., Chichester

  35. Rocha-Arteaga A, Sato K (2003) Topics in infinitely divisible distributions and Lévy processes. Aportaciones Mathemáticas, Investigación 17, Sociedad Matemática Mexicana

  36. Rosiński J (2007) Tempering stable processes. Stoch Process Appl 117(6):677–707

  37. Samorodnitsky G, Taqqu MS (1994) Stable Non-Gaussian random processes: stochastic models with infinite variance. Chapman & Hall, New York

  38. Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge

  39. Schoutens W (2003) Lévy processes in finance: pricing financial derivatives. John Wiley & Sons Ltd., Chichester

  40. Steutal FW, Van Harn K (2004) Infinite divisibility of probability distributions on the real line. Marcel Dekker Inc, New York

  41. Terdik G, Woyczyński WA (2006) Rosiński measures for tempered stable and related Ornstien–Uhlenbeck processes. Probab Math Stat 26(2):213–243

  42. Valdivieso L, Schoutens W, Tuerlinckx F (2009) Maximum likelihood estimation in processes of Ornstein–Uhlenbeck type. Stat Inference Stoch Process 12(1):1–19

  43. Tweedie MCK (1984) An index which distinguishes between some important exponential families. In Ghosh JK, Roy J (eds.) Statistics: applications and new directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Indian Statistical Institute, Calcutta, pp 579–604

  44. Wald A (1949) Note on the consistency of the maximum-likelihood estimate. Ann Math Stat 20(4):595–601

  45. Zhang S, Zhang X (2009) On the transition law of tempered stable Ornstein–Uhlenbeck processes. J Appl Probab 46(3):721–731

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Acknowledgments

The author wishes to thank Professor Gennady Samorodnitsky for the benefit of several discussions and the two anonymous referees whose comments led to improvements in the presentation of this paper.

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Correspondence to Michael Grabchak.

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Grabchak, M. On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes. Stat Inference Stoch Process 19, 29–50 (2016). https://doi.org/10.1007/s11203-015-9118-9

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Keywords

  • Consistent estimator
  • MLE
  • Selfdecomposable distributions
  • Ornstein–Uhlenbeck processes
  • Tempered stable distributions
  • Stable distributions

Mathematics Subject Classification

  • 62F12
  • 62M05
  • 60G51