×

Sharp adaptive estimation of the drift function for ergodic diffusions. (English) Zbl 1084.62079

From the paper: We consider the statistical problem of estimating the drift function of a diffusion process \(X\), given as the solution of the stochastic differential equation \[ dX_t= S(X_t)dt+\sigma(X_t) dW_t,\quad X_0=\xi,\;t\geq 0, \] where \(W\) is a standard Brownian motion and the initial value \(\xi\) is a random variable independent of \(W\). We assume that a continuous record of observations \(X^T=(X_t,0\leq t\leq T)\) is available. The goal is to estimate the function \(S(\cdot)\), which is commonly referred to as the drift function and is interpreted as the instantaneous mean of the process \(X\).
The global estimation problem of the drift function is considered for ergodic diffusion processes. The unknown drift \(S(\cdot)\) is supposed to belong to a nonparametric class of smooth functions of order \(k\geq 1\), but the value of \(k\) is not known to the statistician. A fully data-driven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Aït-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262. · Zbl 1104.62323
[2] Aït-Sahalia, Y. and Mykland, P. (2004). Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 2186–2222. · Zbl 1062.62155
[3] Banon, G. (1978). Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 380–395. · Zbl 0404.93045
[4] Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843–874. · Zbl 1029.62032
[5] Dalalyan, A. S. (2002). Sharp adaptive estimation of the trend coefficient for ergodic diffusion. Prépublication no. 02-1, Université du Maine. Available at www.univ-lemans.fr/sciences/statist/liens/publications.html.
[6] Dalalyan, A. S. and Kutoyants, Yu. A. (2002). Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Methods Statist. 11 402–427.
[7] Delattre, S. and Hoffmann, M. (2002). Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 139–174. · Zbl 1040.60067
[8] Delattre, S., Hoffmann, M. and Kessler, M. (2002). Dynamics adaptive estimation of a scalar diffusion. Prépublication PMA-762, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/.
[9] Efromovich, S. Yu. (1985). Non-parametric estimation of a density with unknown smoothness. Theory Probab. Appl. 30 557–568. · Zbl 0581.62035
[10] Efromovich, S. Yu. (1999). Nonparametric Curve Estimation . Springer, New York. · Zbl 0935.62039
[11] Efromovich, S. Yu. and Pinsker, M. S. (1984). A self-training algorithm for nonparametric filtering. Autom. Remote Control 1984 (11) 58–65.
[12] Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273–1294. · Zbl 0729.62076
[13] Fan, J. (1993). Local linear regression smoothers and their minimax efficiencies. Ann. Statist. 21 196–216. · Zbl 0773.62029
[14] Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics (with discussion). Statist. Sci. 20 317–357. · Zbl 1130.62364
[15] 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
[16] Friedman, A. (1964). Partial Differential Equations of Parabolic Type . Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0144.34903
[17] Galtchouk, L. and Pergamenshchikov, S. (2001). Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes. Math. Methods Statist. 10 316–330. · Zbl 1005.62070
[18] Gihman, I. I. and Skorohod, A. V. (1972). Stochastic Differential Equations . Springer, New York. · Zbl 0242.60003
[19] Golubev, G. K. (1987). Adaptive asymptotically minimax estimates of smooth signals. Problems Inform. Transmission 23 (1) 57–67. · Zbl 0636.94005
[20] Golubev, G. K. (1991). Local asymptotic normality in problems of nonparametric estimation of functions and lower bounds for quadratic risks. Theory Probab. Appl. 36 152–157. · Zbl 0738.62043
[21] Golubev, G. K. (1992). Non-parametric estimation of smooth densities of a distribution in \(L^2\). Problems Inform. Transmission 28 (1) 44–54. · Zbl 0785.62039
[22] Golubev, G. K. and Nussbaum, M. (1992). Adaptive spline estimates in a nonparametric regression model. Theory Probab. Appl. 37 521–529. · Zbl 0787.62044
[23] Hoffmann, M. (1999). Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 135–163. · Zbl 1043.62528
[24] Jacod, J. (2001). Inference for stochastic processes. Prépublication PMA-683, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/.
[25] Jiang, G. J. and Knight, J. (1997). A nonparametric approach to the estimation of diffusion processes—with an application to a short-term interest rate model. Econometric Theory 13 615–645.
[26] Kutoyants, Yu. A. (1998). Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 131–155. · Zbl 0953.62085
[27] Kutoyants, Yu. A. (2004). Statistical Inference for Ergodic Diffusion Processes . Springer, New York. · Zbl 1038.62073
[28] Lepskii, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Theory Probab. Appl. 36 682–697. · Zbl 0776.62039
[29] Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543. · Zbl 1053.62556
[30] Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45 7–38. · Zbl 0719.60089
[31] Pham, T. D. (1981). Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist. Ser. Statist. 12 61–73. · Zbl 0485.62089
[32] Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian noise. Problems Inform. Transmission 16 (2) 52–68. · Zbl 0452.94003
[33] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006
[34] Spokoiny, V. G. (2000). Adaptive drift estimation for nonparametric diffusion model. Ann. Statist. 28 815–836. · Zbl 1105.62330
[35] van Zanten, H. (2001). Rates of convergence and asymptotic normality of kernel estimators for ergodic diffusion processes. J. Nonparametr. Statist. 13 833–850. · Zbl 0999.62030
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.