The nonparametric LAN expansion for discretely observed diffusions

TL;DR

This paper establishes the local asymptotic normality (LAN) property for nonparametric drift estimation in discretely observed reflected diffusions, using PDE techniques and spectral analysis, under mild regularity conditions.

Contribution

It proves the LAN property for nonparametric drift estimation in discretely observed diffusions, connecting stochastic processes with PDE regularity and spectral theory.

Findings

LAN property holds for low frequency sampled diffusions

Spectral analysis of elliptic operators is used in the proof

Regularity estimates from PDE theory are crucial

Abstract

Consider a scalar reflected diffusion $(X_t:t\geq 0)$, 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 $(X_t:t\geq 0)$.