Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations

Abstract

We consider adaptive Bayesian estimation of both drift and diffusion coefficient parameters for ergodic multidimensional diffusion processes based on sampled data. Under a general condition on the discretization step of the sampled data, three kinds of adaptive Bayes type estimators are proposed by applying adaptive maximum likelihood type methods of Uchida and Yoshida (Stoch Process Appl 122:2885–2924, 2012) to Bayesian procedures. We show asymptotic normality and convergence of moments for the adaptive Bayes type estimators by means of the Ibragimov–Has’minskii–Kutoyants program together with the polynomial type large deviation inequality for the statistical random field.