JUMP FILTERING AND EFFICIENT DRIFT ESTIMATION FOR LEVY-DRIVEN SDES

• 成果类型: Article
• 署名作者: Gloter, Arnaud; Loukianova, Dasha; Mai, Hilmar
• 署名单位: Universite Paris Saclay; Institut Polytechnique de Paris; ENSAE Paris
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1591
• 发表日期: 2018
• 页码: 1445-1480
• 关键词: ornstein-uhlenbeck processes; diffusion-processes; high-frequency; models

摘要

The problem of drift estimation for the solution X of a stochastic differential equation with Levy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density, these conditions reduce to n Delta(3-epsilon)(n)-> 0, where n is the number of observations and Delta(n), is the maximal sampling step. This result relaxes the condition n Delta(2)(n)-> 0 usually required for joint estimation of drift and diffusion coefficient for SDEs with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part X-c in the likelihood function. In order to construct the drift estimator, we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to X-c. Convergence results of independent interest are proved for these nonparametric estimators.