Nondegeneracy of Random Field and Estimation of Diffusion

Abstract

We construct a quasi likelihood analysis for diffusions under the high-frequency sampling over a finite time interval. For this, we prove a polynomial type large deviation inequality for the quasi likelihood random field. Then it becomes crucial to prove nondegeneracy of a key index chi_0. By nature of the sampling setting, chi_0 is random. This makes it difficult to apply a naive sufficient condition, and requires a new machinery. In order to establish a quasi likelihood analysis, we need quantitative estimate of the nondegeneracy of chi_0. The existence of a nondegenerate local section of a certain tensor bundle associated with the statistical random field solves this problem.