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Nonparametric estimation of scalar diffusions based on low frequency data. (English) Zbl 1056.62091
Summary: We study the problem of estimating the coefficients of a diffusion \((X_1,t\geq 0)\); the estimation is based on discrete data \(X_{n \Delta}\), \(N=0,1, \dots,N\). The sampling frequency \(\Delta^{-1}\) 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 nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain \((X_{n\Delta}, n=0,1, \dots,N)\) in a suitable Sobolev norm, together with an estimation of its invariant density.

MSC:
62M05 Markov processes: estimation; hidden Markov models
47D07 Markov semigroups and applications to diffusion processes
62G05 Nonparametric estimation
62G99 Nonparametric inference
62M15 Inference from stochastic processes and spectral analysis
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