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Approximate martingale estimating functions for stochastic differential equations with small noises

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  • Uchida, Masayuki

Abstract

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter [epsilon] from discrete time observations at n regularly spaced time points k/n, k=0,1,...,n. We show asymptotic efficiency of an M-estimator derived from the approximate martingale estimating function as [epsilon]-->0 and n-->[infinity] simultaneously.

Suggested Citation

  • Uchida, Masayuki, 2008. "Approximate martingale estimating functions for stochastic differential equations with small noises," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1706-1721, September.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:9:p:1706-1721
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    References listed on IDEAS

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    1. Masayuki Uchida & Nakahiro Yoshida, 2004. "Asymptotic Expansion for Small Diffusions Applied to Option Pricing," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 189-223, October.
    2. Naoto Kunitomo & Akihiko Takahashi, 2001. "The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 117-151, January.
    3. Masayuki Uchida & Nakahiro Yoshida, 2004. "Information Criteria for Small Diffusions via the Theory of Malliavin–Watanabe," Statistical Inference for Stochastic Processes, Springer, vol. 7(1), pages 35-67, March.
    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. Akihiko Takahashi & Nakahiro Yoshida, 2004. "An Asymptotic Expansion Scheme for Optimal Investment Problems," Statistical Inference for Stochastic Processes, Springer, vol. 7(2), pages 153-188, May.
    6. Yoshida, Nakahiro, 1992. "Estimation for diffusion processes from discrete observation," Journal of Multivariate Analysis, Elsevier, vol. 41(2), pages 220-242, May.
    7. Yoshida, Nakahiro, 2003. "Conditional expansions and their applications," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 53-81, September.
    8. Masayuki Uchida, 2004. "Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(4), pages 553-566, December.
    9. Nakahiro Yoshida, 1990. "Asymptotic behavior of M-estimator and related random field for diffusion process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 221-251, June.
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    Cited by:

    1. Ma, Chunhua & Yang, Xu, 2014. "Small noise fluctuations of the CIR model driven by α-stable noises," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 1-11.
    2. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    3. Ren, Panpan & Wu, Jiang-Lun, 2021. "Least squares estimation for path-distribution dependent stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    4. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    5. Yasutaka Shimizu, 2017. "Threshold Estimation for Stochastic Processes with Small Noise," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 951-988, December.
    6. Long, Hongwei & Shimizu, Yasutaka & Sun, Wei, 2013. "Least squares estimators for discretely observed stochastic processes driven by small Lévy noises," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 422-439.
    7. Long, Hongwei, 2009. "Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2076-2085, October.
    8. Uchida, Masayuki & Yoshida, Nakahiro, 2013. "Quasi likelihood analysis of volatility and nondegeneracy of statistical random field," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2851-2876.
    9. Tetsuya Kawai & Masayuki Uchida, 2023. "Adaptive inference for small diffusion processes based on sampled data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(6), pages 643-696, August.
    10. Yang, Xu, 2017. "Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 18-27.
    11. Yusuke Kaino & Masayuki Uchida, 2018. "Hybrid estimators for small diffusion processes based on reduced data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(7), pages 745-773, October.

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