On the LAMN property for continuous observations of some diffusion processes with jumps
On the LAMN property for continuous observations of some diffusion processes with jumps
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关于某些具有跳跃的扩散过程的连续观测的 LAMN 特性
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
E. Nualart
中科院分区：
文献类型：
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作者：
N. Tran;E. Nualart
本文考虑一类具有跳跃的扩散过程，其漂移和跳跃系数依赖于一个未知参数。然后，我们给出了过程在时间区间$[0；T]$内连续观测时局部渐近混合正态(LAMN)性质的一个自包含证明，并由此导出了遍历情形下的局部渐近正态(LAMN)性质。为此，我们给出了一个纯跳跃鞅的Girsanov定理和一个中心极限定理的证明。我们的结果可以看作是Luschgy[15]利用Jacod和Shiryaev[9]中得到的关于半鞅的Girsanov定理以及Sorensen[21]和Feigin[3]建立的关于半鞅的中心极限定理证明了半鞅的LAMN性质的结果。本文的目的是给出这些结果的一个证明，而不是使用抽象的半鞅理论，而是关于Poisson随机测度的积分方程。
In this paper, we consider a diffusion process with jumps whose drift and jump coefficient depend on an unknown parameter. We then give a self-contained proof of the local asymptotic mixed normality (LAMN) property when the process is observed continuously in a time interval $[0; T]$ as $T\to\infty$, and derive, as a consequence, the local asymptotic normality (LAN) property in the ergodic case. For this, we give a proof of a Girsanov's theorem and a Central Limit theorem for a pure jump martingale. Our results could be viewed as a consequence of the LAMN property for semimartingales proved by Luschgy [15], using the Girsanov's theorem for semimartingales obtained in Jacod and Shiryaev [9], and the Central Limit theorem for semimartingales established by Sorensen [21] and Feigin [3]. The aim of this paper is to present a proof of these results without using this abstract semimartingale theory but integral equations with respect to Poisson random measures.