Nonparametric estimation of scalar diffusions based on low frequency data

We study the problem of estimating the coefficients of a diffusion (X t , t ≥ 0); the estimation is based on discrete data X n Δ, 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.

[1]  Jianqing Fan,et al.  A Reexamination of Diffusion Estimators With Applications to Financial Model Validation , 2003 .

[2]  E. Gobet.  LAN property for ergodic diffusions with discrete observations , 2002 .

[3]  Local Asymptotic Normality , 2000 .

[4]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[5]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[6]  Michael Sørensen,et al.  Estimating equations based on eigenfunctions for a discretely observed diffusion process , 1999 .

[7]  Marc Hoffmann,et al.  Adaptive estimation in diffusion processes , 1999 .

[8]  Neil D. Pearson,et al.  Is the Short Rate Drift Actually Nonlinear , 2000 .

[9]  L. Hansen,et al.  Spectral methods for identifying scalar diffusions , 1998 .

[10]  Jianqing Fan,et al.  Efficient Estimation of Conditional Variance Functions in Stochastic Regression , 1998 .

[11]  G. Viennet,et al.  Lp adaptive density estimation in a β mixing framework , 1998 .

[12]  Richard Stanton.  A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk , 1997 .

[13]  R. Bass.  Diffusions and Elliptic Operators , 1997 .

[14]  Mathieu Kessler.  Estimation of an Ergodic Diffusion from Discrete Observations , 1997 .

[15]  H. Selbmann,et al.  Learning to recognize objects , 1999, Trends in Cognitive Sciences.

[16]  Yacine Aït-Sahalia.  Nonparametric Pricing of Interest Rate Derivative Securities , 1995 .

[17]  A. Tsybakov,et al.  Minimax theory of image reconstruction , 1993 .

[18]  N. Yoshida.  Estimation for diffusion processes from discrete observation , 1992 .

[19]  Valentine Genon-Catalot,et al.  Maximnm contrast estimation for diffusion processes from discrete observations , 1990 .

[20]  Ulrich Stadtmüller,et al.  Estimation of Heteroscedasticity in Regression Analysis , 1987 .

[21]  F. Kittaneh.  On Lipschitz functions of normal operators , 1985 .

[22]  N. Krylov,et al.  Statistics and control of stochastic processes , 1985 .

[23]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[24]  P. Tuan.  Nonparametric estimation of the drift coefficient in the diffusion equation , 1981 .

[25]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[26]  G. Banon.  Nonparametric Identification for Diffusion Processes , 1978 .

[27]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[28]  B. M. Brown,et al.  Asymptotic likelihood theory for diffusion processes , 1975, Journal of Applied Probability.

[29]  M. Rosenblatt.  Markov Processes, Structure and Asymptotic Behavior , 1971 .

[30]  S. Varadhan,et al.  Diffusion processes with boundary conditions , 1971 .

[31]  Tosio Kato.  Perturbation theory for linear operators , 1966 .