## 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
Full Text:

### References:

 [1] Aït-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64 527–560. · Zbl 0844.62094 [2] Banon, G. (1978). Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 380–395. · Zbl 0404.93045 [3] Bass, R. F. (1998). Diffusions and Elliptic Operators . Springer, New York. · Zbl 0914.60009 [4] Brown, B. M. and Hewitt, J. I. (1975). Asymptotic likelihood theory for diffusion processes. J. Appl. Probability 12 228–238. · Zbl 0314.62036 [5] Chapman, D. A. and Pearson, N. D. (2000). Is the short rate drift actually nonlinear? J. Finance 55 355–388. [6] Chatelin, F. (1983). Spectral Approximation of Linear Operators . Academic Press, New York. · Zbl 0517.65036 [7] Chen, X., Hansen, L. P. and Scheinkman, J. A. (1997). Shape preserving spectral approximation of diffusions. Working paper. (Last version, November 2000.) [8] Cohen, A. (2000). Wavelet methods in numerical analysis. In Handbook of Numerical Analysis 7 (P. G. Ciarlet, ed.) 417–711. North-Holland, Amsterdam. · Zbl 0976.65124 [9] Davies, E. B. (1995). Spectral Theory and Differential Operators . Cambridge Univ. Press. · Zbl 0893.47004 [10] Engel, K.-J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations . Springer, Berlin. · Zbl 0952.47036 [11] Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660. · Zbl 0918.62065 [12] Fan, J. and Zhang, C. (2003). A re-examination of diffusion estimators with applications to financial model validation. J. Amer. Statist. Assoc. 98 118–134. · Zbl 1073.62571 [13] Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 711–737. · Zbl 1018.60076 [14] Hansen, L. P., Scheinkman, J. A. and Touzi, N. (1998). Spectral methods for identifying scalar diffusions. J. Econometrics 86 1–32. · Zbl 0962.62094 [15] Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 135–163. · Zbl 1043.62528 [16] Kato, T. (1995). Perturbation Theory for Linear Operators . Springer, Berlin. [Reprint of the corrected printing of the 2nd ed. (1980).] · Zbl 0836.47009 [17] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist. 24 211–229. · Zbl 0879.60058 [18] Kessler, M. and Sørensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5 299–314. · Zbl 0980.62074 [19] Kittaneh, F. (1985). On Lipschitz functions of normal operators. Proc. Amer. Math. Soc. 94 416–418. · Zbl 0549.47006 [20] Korostelev, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction . Lecture Notes in Statist. 82 . Springer, Berlin. · Zbl 0833.62039 [21] Kutoyants, Y. A. (1975). Local asymptotic normality for processes of diffusion type. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 10 103–112. [22] Kutoyants, Y. A. (1984). On nonparametric estimation of trend coefficient in a diffusion process. In Statistics and Control of Stochastic Processes 230–250. Nauka, Moscow. · Zbl 0591.62075 [23] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes 1 . General Theory , 2nd ed. Springer, Berlin. · Zbl 1008.62072 [24] Müller, H.-G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610–625. JSTOR: · Zbl 0632.62040 [25] Pham, D. T. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist. Ser. Statist. 12 61–73. · Zbl 0485.62089 [26] Reiß, M. (2003). Simulation results for estimating the diffusion coefficient from discrete time observation. Available at www.mathematik.hu-berlin.de/ reiss/sim-diff-est.pdf. [27] Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior. Springer, Berlin. · Zbl 0236.60002 [28] Stanton, R. (1997). A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 52 1973–2002. [29] Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147–225. · Zbl 0227.76131 [30] Stroock, D. W. and Varadhan, S. R. S. (1997). Multidimensional Diffusion Processes. Springer, Berlin. · Zbl 0426.60069 [31] Tribouley, K. and Viennet, G. (1998). $$L_ p$$ adaptive density estimation in a $$\beta$$-mixing framework. Ann. Inst. H. Poincaré Probab. Statist. 34 179–208. · Zbl 0941.62041 [32] Yoshida, N. (1992). Estimation for diffusion processes from discrete observations. J. Multivariate Anal. 41 220–242. · Zbl 0811.62083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.