The nonparametric LAN expansion for discretely observed diffusions

Abstract

Consider a scalar reflected diffusion $(X_t:t\ge 0)$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of $(X_0,X_{\Delta },...,X_{n\Delta })$ for some fixed sampling distance $\Delta >0$, the model satisfies the local asymptotic normality (LAN) property, assuming that $b$ satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools will be regularity estimates from the theory of parabolic PDEs as well as a detailed analysis of the spectral properties of the elliptic differential operator related to $(X_t:t\ge 0)$.

arXiv comment30 pages

References in corpus (6)

• On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures
• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• Reversible jump MCMC for nonparametric drift estimation for diffusion processes
• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation
• On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems