Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/65321 
Year of Publication: 
2002
Series/Report no.: 
SFB 373 Discussion Paper No. 2002,57
Publisher: 
Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes, Berlin
Abstract: 
We study the problem of estimating the coefficients of a diffusion (Xl, t 2: 0); the estimation is based on discrete data Xn . . n = 0, 1, ... ,N. The sampling frequency delta t is constant , and asymptotics arc taken at the number 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-crror lOBS 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. An important consequence of this result is that we can characterize quantitatively the difference between the estimation of a diffusion in the low frequency sampling case and inference problems in other related frameworks: nonparametric estimation of a diffusion based on continuous or high frequency data: but also parametric estimation for fixed delta, in which case root-N-consistent estimators usually exist. 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/lA, n = 0,1, ... ,N) in a suitable Sobolev norm: together with an estimation of its invariant density.
Subjects: 
Diffusion processes
Nonparametric estimation
Discrete Sarnpling
Low frequency data
Spectral approximation
Ill-posed problems
Persistent Identifier of the first edition: 
Document Type: 
Working Paper

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