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The nonparametric LAN expansion for discretely observed diffusions

6 February 2018
Sven Wang
ArXiv (abs)PDFHTML
Abstract

Consider a scalar reflected diffusion (Xt:t≥0)(X_t:t\geq 0)(Xt​:t≥0), where the unknown drift function bbb is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of (X0,XΔ,...,XnΔ)(X_0,X_\Delta,...,X_{n\Delta})(X0​,XΔ​,...,XnΔ​) for some fixed sampling distance Δ>0\Delta>0Δ>0, the model satisfies the local asymptotic normality (LAN) property, assuming that bbb 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 (Xt:t≥0)(X_t:t\geq 0)(Xt​:t≥0).

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