M-estimation for discretely observed ergodic diffusion processes with infinitely many jumps

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24 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)179-225
Number of pages47
JournalStatistical Inference for Stochastic Processes
Volume9
Issue number2
DOIs
Publication statusPublished - 2006 Jul
Externally publishedYes

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M-estimation
Ergodic Processes
Diffusion Process
Jump
Stochastic Equations
Parametric Inference
Discrete Observations
Differential equation
Random Measure
Estimating Function
Estimating Equation
Asymptotic Normality
Estimator
Interval
Simulation
Model

Keywords

  • Asymptotic normality
  • Diffusion process with jumps
  • Discrete observation
  • Infinitely many jumps
  • M-estimation
  • Parametric inference
  • Partial efficiency

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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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 L{\'e}vy process with a L{\'e}vy 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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AB - 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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KW - M-estimation

KW - Parametric inference

KW - Partial efficiency

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