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Martingale expansion in mixed normal limit

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  • Yoshida, Nakahiro

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 by Yoshida [28] 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”) are discussed.

Suggested Citation

  • Yoshida, Nakahiro, 2013. "Martingale expansion in mixed normal limit," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 887-933.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:3:p:887-933
    DOI: 10.1016/j.spa.2012.10.007
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    References listed on IDEAS

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    1. Yoshida, Nakahiro, 1992. "Estimation for diffusion processes from discrete observation," Journal of Multivariate Analysis, Elsevier, vol. 41(2), pages 220-242, May.
    2. Yuji Sakamoto & Nakahiro Yoshida, 1998. "Asymptotic Expansion of M ‐Estimator Over Wiener Space," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 85-103, January.
    3. Iacus, Stefano Maria & Uchida, Masayuki & Yoshida, Nakahiro, 2009. "Parametric estimation for partially hidden diffusion processes sampled at discrete times," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1580-1600, May.
    4. Yuji Sakamoto & Nakahiro Yoshida, 2004. "Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(3), pages 545-597, September.
    5. Masayuki Uchida, 2010. "Contrast-based information criterion for ergodic diffusion processes from discrete observations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(1), pages 161-187, February.
    6. Masayuki Uchida & Nakahiro Yoshida, 2001. "Information Criteria in Model Selection for Mixing Processes," Statistical Inference for Stochastic Processes, Springer, vol. 4(1), pages 73-98, January.
    7. Masuda, H. & Yoshida, N., 2005. "Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1167-1186, July.
    8. Yasutaka Shimizu & Nakahiro Yoshida, 2006. "Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations," Statistical Inference for Stochastic Processes, Springer, vol. 9(3), pages 227-277, October.
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    Cited by:

    1. Yamagishi, Hayate & Yoshida, Nakahiro, 2023. "Order estimate of functionals related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 490-543.
    2. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    3. Elisa Alòs & Masaaki Fukasawa, 2021. "The asymptotic expansion of the regular discretization error of Itô integrals," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 323-365, January.
    4. Podolskij, Mark & Veliyev, Bezirgen & Yoshida, Nakahiro, 2017. "Edgeworth expansion for the pre-averaging estimator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3558-3595.
    5. Ulrich Hounyo & Bezirgen Veliyev, 2016. "Validity of Edgeworth expansions for realized volatility estimators," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
    6. Nakahiro Yoshida, 2022. "Quasi-likelihood analysis and its applications," Statistical Inference for Stochastic Processes, Springer, vol. 25(1), pages 43-60, April.
    7. Tudor, Ciprian A. & Yoshida, Nakahiro, 2019. "Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3499-3526.
    8. Yamagishi, Hayate & Yoshida, Nakahiro, 2024. "Asymptotic expansion of the quadratic variation of fractional stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    9. Tudor, Ciprian A. & Yoshida, Nakahiro, 2023. "High order asymptotic expansion for Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 443-492.

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