Nonparametric estimation of scalar diffusions based on low frequency data

Abstract

We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{nΔ},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 (X_{nΔ},n=0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invariant density.

Published at http://dx.doi.org/10.1214/009053604000000797 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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