Nonparametric estimation of scalar diffusions based on low frequency data

Emmanuel Gobet, Marc Hoffmann, Markus Reiß

The Annals of Statistics · 2004 · Volume 32 · Issue 5

Abstract

We study the problem of estimating the coefficients of a diffusion (Xt,t≥0); the estimation is based on discrete data XnΔ,n=0,1,…,N. The sampling frequency Δ−1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue–eigenfunction pair of the transition operator of the discrete time Markov chain (XnΔ,n=0,1,…,N) in a suitable Sobolev norm, together with an estimation of its invariant density.

Keywords

62G99, 62M05, 62M15, Diffusion processes, discrete sampling, Ill-posed problems, low frequency data, nonparametric estimation, spectral approximation

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