Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes

We consider nonparametric estimation of the Levy measure of a hidden Levy process driving a stationary Omstein-Uhlenbeck process which is observed at discrete time points. This Levy measure can be expressed in terms of the canonical function of the stationary distribution of the Omstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a suppport-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Oenstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included. © 2005 ISI/BS.

[1]  G. Jongbloed,et al.  Parametric Estimation for Subordinators and Induced OU Processes , 2006 .

[2]  J. Wellner,et al.  The suppport reduction algorithm for computing nonparametric function estimates in mixture models , 2004, math/0405511.

[3]  G. Roberts,et al.  Bayesian inference for non‐Gaussian Ornstein–Uhlenbeck stochastic volatility processes , 2004 .

[4]  Hiroki Masuda.  On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process , 2004 .

[5]  F. Steutel,et al.  Infinite Divisibility of Probability Distributions on the Real Line , 2003 .

[6]  Thierry Jeantheau,et al.  Stochastic volatility models as hidden Markov models and statistical applications , 2000 .

[7]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[8]  N. Shephard,et al.  Likelihood analysis of a first‐order autoregressive model with exponential innovations , 1999 .

[9]  P. Hughett.  Error Bounds for Numerical Inversion of a Probability Characteristic Function , 1998 .

[10]  S. Satchell,et al.  The Cumulant Generating Function Estimation Method , 1997, Econometric Theory.

[11]  Avishai Mandelbaum,et al.  On Harris Recurrence in Continuous Time , 1994, Math. Oper. Res..

[12]  S. Meyn,et al.  Stability of Markovian processes II: continuous-time processes and sampled chains , 1993, Advances in Applied Probability.

[13]  T. Shiga,et al.  A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type , 1990 .

[14]  M. Thompson,et al.  Minimum‐distance methods based on quadratic distances for transforms , 1987 .

[15]  A. Feuerverger,et al.  On the Efficiency of Empirical Characteristic Function Procedures , 1981 .

[16]  Y. Davydov.  Mixing Conditions for Markov Chains , 1974 .

[17]  B. Schorr,et al.  Numerical inversion of a class of characteristic functions , 1973 .

[18]  Erhan Çinlar,et al.  A stochastic integral in storage theory , 1971 .

[19]  H. Tucker,et al.  A Graduate Course in Probability , 1968 .

[20]  H. Rubin,et al.  Estimating the Parameters of a Differential Process , 1959 .

[21]  Prof. Martino Grasselli.  Stochastic volatility models , 2007 .

[22]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[23]  E. Rio,et al.  Théorie asymptotique de processus aléatoires faiblement dépendants , 2000 .

[24]  N. Shephard,et al.  Modelling by L´ evy Processes for Financial Econometrics , 2000 .

[25]  C. LareÂdo,et al.  Stochastic volatility models as hidden Markov models and statistical applications , 2000 .

[26]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[27]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[28]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Notes and comments , 1984 .

[29]  P. Brockwell,et al.  Non‐Parametric Estimation for Non‐Decreasing Lévy Processes , 1982 .

[30]  R. Getoor,et al.  Markov Processes: Ray Processes and Right Processes , 1975 .

[31]  G. Jameson.  Topology and Normed Spaces , 1974 .

[32]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .