The nonparametric LAN expansion for discretely observed diffusions

Abstract

Consider a scalar reflected diffusion (Xt : t ≥ 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 ∆ , ..., X n∆ ) for some fixed sampling distance ∆ > 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 used are regularity estimates for certain parabolic PDEs as well as a detailed analysis of the spectral properties of the elliptic differential operator related to (Xt : t ≥ 0). MSC 2010 subject classifications: 62M99. 1329 1330 S. Wang LAN expansion for discretely observed diffusions 1331• The second main ingredient consists of two well known limit theorems, the central limit theorem for martingale difference sequences [4] and the ergodic theorem, which ensure the right limits for the first and second order terms in the Taylor expansion respectively.