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Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes

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  • Clément, Emmanuelle
  • Gloter, Arnaud

Abstract

We prove the Local Asymptotic Mixed Normality property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process. The process is observed on the fixed time interval [0,1] and the parameter appears in the drift coefficient only. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof of this result contains a sharp study of the asymptotic behavior, in small time, of the transition probability density of the process and of its logarithm derivative.

Suggested Citation

  • Clément, Emmanuelle & Gloter, Arnaud, 2015. "Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2316-2352.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:6:p:2316-2352
    DOI: 10.1016/j.spa.2015.01.002
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    References listed on IDEAS

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    1. Kawai, Reiichiro & Masuda, Hiroki, 2011. "On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 460-469, April.
    2. Yacine Aït-Sahalia & Jean Jacod, 2008. "Fisher's Information for Discretely Sampled Lévy Processes," Econometrica, Econometric Society, vol. 76(4), pages 727-761, July.
    3. Ishikawa, Yasushi & Kunita, Hiroshi, 2006. "Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1743-1769, December.
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    Cited by:

    1. Jakobsen, Nina Munkholt & Sørensen, Michael, 2019. "Estimating functions for jump–diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3282-3318.
    2. Masuda, Hiroki, 2019. "Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 1013-1059.

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