Skip to main content
SpringerLink
Log in

Asymptotic Expansions for Stochastic Processes

  • Chapter
  • First Online:
Rabi N. Bhattacharya

Part of the book series: Contemporary Mathematicians ((CM))

Abstract

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived in Yoshida 1997 as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The order of the central limit theorem is referred to as the first order in asymptotic decision theory, differently from the numbering of terms in asymptotic expansion.

References

  1. Shun-ichi Amari. Differential-geometrical Methods in Statistics, volume 28 of Lecture Notes in Statistics. Springer-Verlag, New York, 1985.

    Google Scholar 

  2. Torben G. Andersen and Tim Bollerslev. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39:885–905, 1998.

    Google Scholar 

  3. Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, and Paul Labys. The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc., 96(453):42–55, 2001.

    Google Scholar 

  4. Theodore W. Anderson. On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann. Math. Statist., 30:676–687, 1959.

    Google Scholar 

  5. Ole E. Barndorff-Nielsen and Neil Shephard. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B Stat. Methodol., 64(2):253–280, 2002.

    Google Scholar 

  6. Ole E. Barndorff-Nielsen and Neil Shephard. Econometric analysis of realized covariation: high frequency based covariance, regression, and correlation in financial economics. Econometrica, 72(3):885–925, 2004.

    Google Scholar 

  7. Ishwar V. Basawa and Hira L. Koul. Asymptotic tests of composite hypotheses for nonergodic type stochastic processes. Stochastic Process. Appl., 9(3):291–305, 1979.

    Google Scholar 

  8. Ishwar V. Basawa and B. L. S. Prakasa Rao. Statistical Inference for Stochastic Processes. Probability and Mathematical Statistics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1980.

    Google Scholar 

  9. Ishwar V. Basawa and David J. Scott. Asymptotic Optimal Inference for Nonergodic Models, volume 17 of Lecture Notes in Statistics. Springer-Verlag, New York, 1983.

    Google Scholar 

  10. Vidmantas K. Bentkus, Friedrich Götze, and Alexander N. Tikhomirov. Berry-Esseen bounds for statistics of weakly dependent samples. Bernoulli, 3:329–349, 1997.

    Google Scholar 

  11. Rabi Bhattacharya and Manfred Denker. Asymptotic Statistics, volume 14 of DMV Seminar. Birkhäuser, Basel, 1990.

    Google Scholar 

  12. Rabi Bhattacharya and Aramian Wasielak. On the speed of convergence of multidimensional diffusions to equilibrium. Stochastics and Dynamics, 12(1), 2012.

    Google Scholar 

  13. Rabi N. Bhattacharya. Rates of weak convergence for the multidimensional central limit theorem. Teor. Verojatnost. i Primenen, 15:69–85, 1970.

    Google Scholar 

  14. Rabi N. Bhattacharya. Rates of weak convergence and asymptotic expansions for classical central limit theorems. Ann. Math. Statist., 42:241–259, 1971.

    Google Scholar 

  15. Rabi N. Bhattacharya. Recent results on refinements of the central limit theorem. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pages 453–484. Univ. California Press, Berkeley, Calif., 1972.

    Google Scholar 

  16. Rabi N. Bhattacharya. On classical limit theorems for diffusions. Sankhyā (Statistics). The Indian Journal of Statistics. Series A, 44(1):47–71, 1982.

    Google Scholar 

  17. Rabi N. Bhattacharya and Jayanta K. Ghosh. On the validity of the formal Edgeworth expansion. Ann. Statist., 6(2):434–451, 1978.

    Google Scholar 

  18. Rabi N. Bhattacharya and Sundareswaran Ramasubramanian. Recurrence and ergodicity of diffusions. Journal of Multivariate Analysis, 12(1):95–122, 1982.

    Google Scholar 

  19. Rabi N. Bhattacharya and R. Ranga Rao. Normal Approximation and Asymptotic Epansions. Robert E. Krieger Publishing Co. Inc., Melbourne, FL, 1986. Reprint of the 1976 original.

    Google Scholar 

  20. Rabi N. Bhattacharya and R. Ranga Rao. Normal Approximation and Asymptotic Expansions, volume 64 of Classics in Applied Mathematics. SIAM, 2010.

    Google Scholar 

  21. Bo Martin Bibby and Michael Sørensen. Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1(1–2):17–39, 1995.

    Google Scholar 

  22. Klaus Bichteler, Jean-Bernard Gravereaux, and Jean Jacod. Malliavin Calculus for Processes with Jumps, volume 2 of Stochastics Monographs. Gordon and Breach Science Publishers, New York, 1987.

    Google Scholar 

  23. Erwin Bolthausen. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. verw. Gebiete, 54(1):59–73, 1980.

    Google Scholar 

  24. Erwin Bolthausen. Exact convergence rates in some martingale central limit theorems. Ann. Probab., 10(3):672–688, 1982.

    Google Scholar 

  25. Bruce M. Brown. Martingale central limit theorems. Ann. Math. Statist., 42:59–66, 1971.

    Google Scholar 

  26. Harald Cramér. Random Variables and Probability Distributions, volume 36 of Cambridge Tracts in Mathematics. Cambridge University Press,Cambridge, 3rd edition, 1970.

    Google Scholar 

  27. Didier Dacunha-Castelle and Danielle Florens-Zmirou. Estimation of the coefficients of a diffusion from discrete observations. Stochastics, 19(4):263–284, 1986.

    Google Scholar 

  28. Ratan Dasgupta. Nonuniform speed of convergence to normality for some stationary m-dependent processes. Calcutta Statist. Assoc. Bull., 42(167–168):149–162, 1992.

    Google Scholar 

  29. Paul Doukhan. Mixing: Properties and Examples., volume 85 of Lecture Notes in Statistics. Springer-Verlag, New York, 1994.

    Google Scholar 

  30. Aryeh Dvoretsky. Asymptotic normality for sums of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, pages 513–535. Univ. California Press, Berkeley, Calif., 1972.

    Google Scholar 

  31. Paul D. Feigin. Stable convergence of semimartingales. Stochastic Process. Appl., 19(1):125–134, 1985.

    Google Scholar 

  32. Daniéle Florens-Zmirou. Approximate discrete-time schemes for statistics of diffusion processes. Statistics, 20(4):547–557, 1989.

    Google Scholar 

  33. Valentine Genon-Catalot and Jean Jacod. On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 29(1):119–151, 1993.

    Google Scholar 

  34. Friedrich Götze and Christian Hipp. Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrsch. verw. Gebiete, 64(2):211–239, 1983.

    Google Scholar 

  35. Friedrich Götze and Christian Hipp. Asymptotic distribution of statistics in time series. Ann. Statist., 22(4):2062–2088, 1994.

    Google Scholar 

  36. Peter Hall. The Bootstrap and Edgeworth Expansion. Springer Series in Statistics. Springer-Verlag, New York, 1992.

    Google Scholar 

  37. Peter Hall and Christopher C. Heyde. Martingale Limit Theory and its Application. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. Probability and Mathematical Statistics.

    Google Scholar 

  38. Erich Häusler. On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab., 16(1):275–299, 1988.

    Google Scholar 

  39. Ildar A. Ibragimov. Some limit theorems for stationary processes. Theory of Probability & Its Applications, 7(4):349–382, 1962.

    Google Scholar 

  40. Nobuyuki Ikeda and Shinzo Watanabe. Stochastic Differential Equations and Diffusion Processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 2nd edition, 1989.

    Google Scholar 

  41. Jean Jacod. On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités, XXXI, volume 1655 of Lecture Notes in Math., pages 232–246. Springer, Berlin, 1997.

    Google Scholar 

  42. Jean Jacod, Andrzej Klopotowski, and Jean Mémin. Théorème de la limite centrale et convergence fonctionnelle vers un processus à accroissements indépendants: la méthode des martingales. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 18(1):1–45, 1982.

    Google Scholar 

  43. Jean Jacod and Albert N. Shiryaev. Limit Theorems for Stochastic Processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003.

    Google Scholar 

  44. Adam Jakubowski, Jean Mémin, and Gilles Pagès. Convergence en loi des suites d’intégrales stochastiques sur l’espace D 1 de Skorokhod. Probability Theory and Related Fields, 81(1):111–137, 1989.

    Google Scholar 

  45. P. Jeganathan. On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā Ser. A, 44(2):173–212, 1982.

    Google Scholar 

  46. Niels Keiding. Correction to: “Estimation in the birth process” (Biometrika 61 (1974), 71–80). Biometrika, 61:647, 1974.

    Google Scholar 

  47. Niels Keiding. Maximum likelihood estimation in the birth-and-death process. Ann. Statist., 3:363–372, 1975.

    Google Scholar 

  48. Mathieu Kessler. Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24(2):211–229, 1997.

    Google Scholar 

  49. Fumiyasu Komaki. On asymptotic properties of predictive distributions. Biometrika, 83(2):299–313, 1996.

    Google Scholar 

  50. Sadanori Konishi and Genshiro Kitagawa. Generalised information criteria in model selection. Biometrika, 83(4):875–890, 1996.

    Google Scholar 

  51. Uwe Küchler and Michael Sørensen. Exponential Families of Stochastic Processes. Springer Series in Statistics. Springer-Verlag, New York, 1997.

    Google Scholar 

  52. Thomas G. Kurtz and Philip Protter. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab., 19(3):1035–1070, 1991.

    Google Scholar 

  53. Shigeo Kusuoka and Nakahiro Yoshida. Malliavin calculus, geometric mixing, and expansion of diffusion functionals. Probab. Theory Related Fields, 116(4):457–484, 2000.

    Google Scholar 

  54. Yury A. Kutoyants. Parameter Estimation for Stochastic Processes, volume 6 of Research and Exposition in Mathematics. Heldermann Verlag, Berlin, 1984. Translated from the Russian and edited by B. L. S. Prakasa Rao.

    Google Scholar 

  55. Yury A. Kutoyants. Statistical Inference for Spatial Poisson Processes, volume 134 of Lecture Notes in Statistics. Springer-Verlag, New York, 1998.

    Google Scholar 

  56. Yury A. Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics. Springer-Verlag London Ltd., London, 2004.

    Google Scholar 

  57. Soumendra Nath Lahiri. Refinements in asymptotic expansions for sums of weakly dependent random vectors. Ann. Probab., 21(2):791–799, 1993.

    Google Scholar 

  58. Robert S. Liptser and Albert N. Shiryaev. On the rate of convergence in the central limit theorem for semimartingales. Theory of Probability & Its Applications, 27(1):1–13, 1982.

    Google Scholar 

  59. Robert S. Liptser and Albert N. Shiryayev. A functional central limit theorem for semimartingales. Theory of Probability & Its Applications, 25(4):667–688, 1981.

    Google Scholar 

  60. Robert S. Liptser and Albert N. Shiryayev. Theory of Martingales, volume 49 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the 1974 Russian original by A.B. Aries.

    Google Scholar 

  61. Hiroki Masuda. Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl., 117(1):35–56, 2007.

    Google Scholar 

  62. Hiroki Masuda et al. Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann. Statist., 41(3):1593–1641, 2013.

    Google Scholar 

  63. Donald L. McLeish. Dependent central limit theorems and invariance principles. Ann. Probab., 2:620–628, 1974.

    Google Scholar 

  64. Sean P. Meyn and Richard L. Tweedie. Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Probab., 24(3):542–574, 1992.

    Google Scholar 

  65. Sean P. Meyn and Richard L. Tweedie. Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Probab., 25(3):487–517, 1993.

    Google Scholar 

  66. Sean P. Meyn and Richard L. Tweedie. Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25(3):518–548, 1993.

    Google Scholar 

  67. Sean P. Meyn and Richard L. Tweedie. Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge, 2nd edition, 2009. (With a prologue by Peter W. Glynn.).

    Google Scholar 

  68. Per Aslak Mykland. Asymptotic expansions and bootstrapping distributions for dependent variables: a martingale approach. Ann. Statist., 20(2):623–654, 1992.

    Google Scholar 

  69. Per Aslak Mykland. Asymptotic expansions for martingales. Ann. Probab., 21(2):800–818, 1993.

    Google Scholar 

  70. Per Aslak Mykland. Martingale expansions and second order inference. Ann. Statist., 23(3):707–731, 1995.

    Google Scholar 

  71. Sergey V. Nagaev. Some limit theorems for stationary Markov chains. Theory of Probability & Its Applications, 2(4):378–406, 1957.

    Google Scholar 

  72. Sergey V. Nagaev. More exact statement of limit theorems for homogeneous Markov chains. Theory of Probability & Its Applications, 6(1):62–81, 1961.

    Google Scholar 

  73. Teppei Ogihara and N Yoshida. Quasi-likelihood analysis for the stochastic differential equation with jumps. Statistical Inference for Stochastic Processes, 14(3): 189–229, 2011.

    Google Scholar 

  74. Teppei Ogihara and Nakahiro Yoshida. Quasi-likelihood analysis for stochastic regression models with nonsynchronous observations. arXiv preprint; arXiv:1212.4911, 2012.

    Google Scholar 

  75. Valentin V. Petrov. Sums of Independent Random Variables, volume 82 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown.

    Google Scholar 

  76. Mark Podolskij and Nakahiro Yoshida. Edgeworth expansion for functionals of continuous diffusion processes. arXiv preprint; arXiv:1309.2071, 2013.

    Google Scholar 

  77. B. L. S. Prakasa Rao. Asymptotic theory for nonlinear least squares estimator for diffusion processes. Math. Operationsforsch. Statist. Ser. Statist., 14(2):195–209, 1983.

    Google Scholar 

  78. B. L. S. Prakasa Rao. Statistical inference from sampled data for stochastic processes. In Statistical Inference from Stochastic Processes (Ithaca, NY, 1987), volume 80 of Contemp. Math., pages 249–284. Amer. Math. Soc., Providence, RI, 1988.

    Google Scholar 

  79. B.L.S. Prakasa Rao. Semimartingales and their Statistical Inference, volume 83 of Monographs on Statistics and Applied Probability. Chapman & Hall/CRC, Boca Raton, Fl, 1999.

    Google Scholar 

  80. B.L.S. Prakasa Rao. Statistical Inference for Diffusion Type Processes, volume 8 of Kendall’s Library in Statistics. E. Arnold, London; Oxford Univ. Press, New York, 1999.

    Google Scholar 

  81. Malempati M. Rao. Consistency and limit distributions of estimators of parameters in explosive stochastic difference equations. Ann. Math. Statist., 32:195–218, 1961.

    Google Scholar 

  82. Rolando Rebolledo. Central limit theorems for local martingales. Probability Theory and Related Fields, 51(3):269–286, 1980.

    Google Scholar 

  83. Yuji Sakamoto and Nakahiro Yoshida. Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property. Ann. Inst. Statist. Math., 56(3):545–597, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  84. Yuji Sakamoto and Nakahiro Yoshida. Third-order asymptotic expansion of M-estimators for diffusion processes. Ann. Inst. Statist. Math., 61(3):629–661, 2009.

    Google Scholar 

  85. Yasutaka Shimizu and Nakahiro Yoshida. Estimation of parameters for diffusion processes with jumps from discrete observations. Stat. Inference Stoch. Process., 9(3):227–277, 2006.

    Google Scholar 

  86. Charles Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, volume II: Probability Theory, pages 583–602. Univ. California Press, Berkeley, Calif., 1972.

    Google Scholar 

  87. Jonas K. Sunklodas. A lower bound for the rate of convergence in the central limit theorem for m-dependent random fields. Teor. Veroyatnost. i Primenen., 43(1):171–179, 1998.

    Google Scholar 

  88. Trevor J. Sweeting. Speeds of convergence for the multidimensional central limit theorem. Ann. Probab., 5(1):28–41, 1977.

    Google Scholar 

  89. Alexander N. Tikhomirov. On the convergence rate in the central limit theorem for weakly dependent random variables. Theory of Probability & Its Applications, 25(4):790–809, 1981.

    Google Scholar 

  90. Masayuki Uchida. Contrast-based information criterion for ergodic diffusion processes from discrete observations. Annals of the Institute of Statistical Mathematics, 62(1):161–187, 2010.

    Google Scholar 

  91. Masayuki Uchida and Nakahiro Yoshida. Information criteria in model selection for mixing processes. Stat. Inference Stoch. Process., 4(1):73–98, 2001.

    Google Scholar 

  92. Masayuki Uchida and Nakahiro Yoshida. Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Processes and their Applications, 122(8):2885–2924, 2012.

    Google Scholar 

  93. Masayuki Uchida and Nakahiro Yoshida. Quasi likelihood analysis of volatility and nondegeneracy of statistical random field. Stochastic Processes and their Applications, 123(7):2851–2876, 2013.

    Google Scholar 

  94. Shinzo Watanabe. Lectures on Stochastic Differential Equations and Malliavin Calculus. (Notes by Nair, M. Gopalan and Rajeev, B.), volume 73 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Springer-Verlag, Berlin, 1984.

    Google Scholar 

  95. John S. White. The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Statist., 29:1188–1197, 1958.

    Google Scholar 

  96. Nakahiro Yoshida. Estimation for diffusion processes from discrete observation. J. Multivariate Anal., 41(2):220–242, 1992.

    Google Scholar 

  97. Nakahiro Yoshida. Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields, 109(3):301–342, 1997.

    Google Scholar 

  98. Nakahiro Yoshida. Malliavin calculus and martingale expansion. Bull. Sci. Math., 125(6–7):431–456, 2001. Rencontre Franco-Japonaise de Probabilités (Paris, 2000).

    Google Scholar 

  99. Nakahiro Yoshida. Partial mixing and conditional Edgeworth expansion for diffusions with jumps. Probab. Theory Related Fields, 129:559–624, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  100. Nakahiro Yoshida. Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Annals of the Institute of Statistical Mathematics, 63(3):431–479, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  101. Nakahiro Yoshida. Asymptotic expansion for the quadratic form of the diffusion process. arXiv preprint; arXiv:1212.5845, 2012.

    Google Scholar 

  102. Nakahiro Yoshida. Martingale expansion in mixed normal limit. arXiv preprint; arXiv:1210.3680v3, 2012.

    Google Scholar 

Download references

Acknowledgements

This work was in part supported by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 24340015 (Scientific Research), Nos. 24650148 and 26540011 (Challenging Exploratory Research); CREST Japan Science and Technology Agency; and by a Cooperative Research Program of the Institute of Statistical Mathematics.

Author information

Authors and Affiliations

  1. University of Tokyo & Japan Science and Technology Agency, CREST, Tokyo, Japan

    Nakahiro Yoshida

Authors
  1. Nakahiro Yoshida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nakahiro Yoshida .

Editor information

Editors and Affiliations

  1. Mathematics, The Pennsylvania State University, State College, Pennsylvania, USA

    Manfred Denker

  2. Department of Mathematics, Oregon State University, Corvallis, Oregon, USA

    Edward C. Waymire

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Yoshida, N. (2016). Asymptotic Expansions for Stochastic Processes. In: Denker, M., Waymire, E. (eds) Rabi N. Bhattacharya. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30190-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation