Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations

Abstract

The estimate of the probability of the large deviation or the statistical randomfieldisthekeytoensuretheconvergenceofmomentsoftheassociatedestima- tor, and it also plays an essential role to prove mathematical validity of the asymptotic expansion of the estimator. For non-linear stochastic processes, it involves technical difficultiestoshowastandardexponentialtypeestimate;besides,itisnotnecessaryfor these purposes. In this paper, we propose a polynomial-type large deviation inequality which is easily verified by the L p -boundedness of certain functionals; usually they are simple additive functionals. We treat a statistical random field with multi-grades and discuss M and Bayesian type estimators. As an application, we show the behavior of those estimators, including convergence of moments, for the statistical random field in the quasi-likelihood analysis of the stochastic differential equation that is possibly multi-dimensional and non-linear. The results are new even for stochastic differential equations, while they obviously apply to other various statistical models.