### 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 language | English |
---|---|

Pages (from-to) | 179-225 |

Number of pages | 47 |

Journal | Statistical Inference for Stochastic Processes |

Volume | 9 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 Jul |

Externally published | Yes |

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### 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

**M-estimation for discretely observed ergodic diffusion processes with infinitely many jumps.** / Shimizu, Yasutaka.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - M-estimation for discretely observed ergodic diffusion processes with infinitely many jumps

AU - Shimizu, Yasutaka

PY - 2006/7

Y1 - 2006/7

N2 - 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.

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.

KW - Asymptotic normality

KW - Diffusion process with jumps

KW - Discrete observation

KW - Infinitely many jumps

KW - M-estimation

KW - Parametric inference

KW - Partial efficiency

UR - http://www.scopus.com/inward/record.url?scp=33745617674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745617674&partnerID=8YFLogxK

U2 - 10.1007/s11203-005-8113-y

DO - 10.1007/s11203-005-8113-y

M3 - Article

VL - 9

SP - 179

EP - 225

JO - Statistical Inference for Stochastic Processes

JF - Statistical Inference for Stochastic Processes

SN - 1387-0874

IS - 2

ER -