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M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

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Abstract

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f θ(z)dz, and we admit the case ∫ f θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models.

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References

  1. M. G. Akritas R. A. Johnson (1981) ArticleTitleAsymptotic inference in Levy processes of the discontinuous type Ann. Statist. 9 IssueID3 604–614 Occurrence Handle0481.62069 Occurrence Handle615436

    MATH  MathSciNet  Google Scholar 

  2. Bichteler, K. and Jacod, J.: Calcul de Malliavin pour les diffusions avec sauts: existence d’une densite dans le cas unidimensionnel, Lecture Notes in Math. (1986), 132–157. Springer-Verlag, Berlin.

  3. V. Genon-Catalot J. Jacod (1993) ArticleTitleOn the estimation of the diffusion coefficient for multidimensional diffusion process Ann. Inst. Henri Poincaré Prob. Statist. 29 119–151 Occurrence Handle0770.62070 Occurrence Handle1204521

    MATH  MathSciNet  Google Scholar 

  4. I. A. Ibragimov R. Z. Has’minskii (1981) Statistical Estimation Springer-Verlag Berlin Occurrence Handle0467.62026

    MATH  Google Scholar 

  5. J. Jacod A. N. Shiryaev (1987) Limit Theorems for Stochastic Processes Springer-Verlag Berlin Occurrence Handle0635.60021

    MATH  Google Scholar 

  6. M. Kessler (1997) ArticleTitleEstimation of diffusion processes from discrete observations Scand. J. Statist. 24 211–229 Occurrence Handle0879.60058 Occurrence Handle1455868 Occurrence Handle10.1111/1467-9469.00059

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. Kwon (1997) ArticleTitleStability in distribution for a class of diffusion with jump Bull. Korean Math. Soc. 34 IssueID2 287–293 Occurrence Handle0885.60076 Occurrence Handle1455448

    MATH  MathSciNet  Google Scholar 

  8. Y. Kwon C. Lee (1999) ArticleTitleStrong Feller property and irreducibility of diffusions with jumps Stoch. Stoch. Rep. 67 147–157 Occurrence Handle0936.60070 Occurrence Handle1717795

    MATH  MathSciNet  Google Scholar 

  9. H. Masuda (2004) ArticleTitleOn multi-dimensional Ornstein-Uhlenbeck processes driven by a general Lévy process Bernoulli. 10 97–120 Occurrence Handle1048.60060 Occurrence Handle2044595 Occurrence Handle10.3150/bj/1077544605

    Article  MATH  MathSciNet  Google Scholar 

  10. Masuda, H.: Ergodicity and exponential β-mixing bounds for a strong solution of Lévy-driven stochastic differential equations, MHF preprint series, Kyushu University, (2004), 2004–2019.

  11. S.P. Meyn R.L. Tweedie (1993) ArticleTitleStability of Markov processes II: continuous-time processes and sampled chains Adv. Appl. Probab. 25 487–517 Occurrence Handle0781.60052 Occurrence Handle1234294 Occurrence Handle10.2307/1427521

    Article  MATH  MathSciNet  Google Scholar 

  12. S.P. Meyn R.L. Tweedie (1993) ArticleTitleStability of Markov processes III: Foster–Lyapunov criteria for continuous-time processes Adv. Appl. Probab. 24 518–548 Occurrence Handle1234295 Occurrence Handle10.2307/1427522

    Article  MathSciNet  Google Scholar 

  13. Sørensen, M.: A note on the existence of a consistent maximum likelihood estimator for diffusions with jumps, In: Markov Process and Control Theory, Akademie-Verlag, Berlin, 1989, pp. 229–234.

  14. M. Sørensen (1991) Likelihood methods for diffusions with jumps, Statistical Inference in Stochastic Processes Marcel Dekker New York 67–105

    Google Scholar 

  15. Shimizu, Y. and Yoshida, N.: Estimation for diffusion processes with jumps from discrete observations, preprint, submitted (2002).

  16. Woerner, J.: Statistical analysis for discretely observed Lévy processes, Doctor thesis. (2001)

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  1. Division of Mathematical Science, Graduate School of Engineering Science, Osaka University Toyonaka, Osaka, 560-8531, Japan

    Yasutaka Shimizu

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  1. Yasutaka Shimizu
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Correspondence to Yasutaka Shimizu.

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Final version 25 December 2004

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Shimizu, Y. M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps. Stat Infer Stoch Process 9, 179–225 (2006). https://doi.org/10.1007/s11203-005-8113-y

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  • DOI: https://doi.org/10.1007/s11203-005-8113-y

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