Fisher's Information for Discretely Sampled Lévy Processes

This paper studies the asymptotic behavior of Fisher's information for a Levy process discretely sampled at an increasing frequency. As a result, we derive the optimal rates of convergence of efficient estimators of the different parameters of the process and show that the rates are often nonstandard and differ across parameters. We also show that it is possible to distinguish the continuous part of the process from its jumps part, and even different types of jumps from one another. Copyright Copyright 2008 by The Econometric Society.

[1]  W. DuMouchel Stable Distributions in Statistical Inference: 2. Information from Stably Distributed Samples , 1975 .

[2]  C. Klüppelberg,et al.  A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour , 2004, Journal of Applied Probability.

[3]  E. Fama,et al.  Parameter Estimates for Symmetric Stable Distributions , 1971 .

[4]  J. Nolan,et al.  Maximum likelihood estimation and diagnostics for stable distributions , 2001 .

[5]  W. DuMouchel Stable Distributions in Statistical Inference: 1. Symmetric Stable Distributions Compared to other Symmetric Long-Tailed Distributions , 1973 .

[6]  V. Zolotarev One-dimensional stable distributions , 1986 .

[7]  Ioannis A. Koutrouvelis,et al.  Regression-Type Estimation of the Parameters of Stable Laws , 1980 .

[8]  A. Fenech Asymptotically Efficient Estimation of Location for a Symmetric Stable Law , 1976 .

[9]  J. Huston McCulloch,et al.  Measuring Tail Thickness to Estimate the Stable Index α: A Critique , 1997 .

[10]  J. L. Nolan,et al.  Numerical calculation of stable densities and distribution functions: Heavy tails and highly volatil , 1997 .

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

[12]  T. Chan Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[13]  Francis X. Diebold,et al.  Modeling and Forecasting Realized Volatility , 2001 .

[14]  J. Jacod,et al.  Some Remarks on the Joint Estimation of the Index and the Scale Parameter for Stable Processes , 1994 .

[15]  Vedat Akgiray,et al.  Estimation of Stable-Law Parameters: A Comparative Study , 1989 .

[16]  H. Bergström,et al.  On some expansions of stable distribution functions , 1952 .

[17]  K. Singleton Estimation of affine asset pricing models using the empirical characteristic function , 2001 .

[18]  M. Yor,et al.  Stochastic Volatility for Lévy Processes , 2003 .

[19]  S. Rachev,et al.  Modeling asset returns with alternative stable distributions , 1993 .

[20]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[21]  Sidney I. Resnick,et al.  A Simple Asymptotic Estimate for the Index of a Stable Distribution , 1980 .

[22]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[23]  W. DuMouchel On the Asymptotic Normality of the Maximum-Likelihood Estimate when Sampling from a Stable Distribution , 1973 .

[24]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[25]  P. Mykland,et al.  ANOVA for diffusions and Itô processes , 2006, math/0611274.

[26]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[27]  O. Barndorff-Nielsen Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[28]  S. Levendorskii,et al.  Perpetual American options under Levy processes , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[29]  Ernesto Mordecki,et al.  Optimal stopping and perpetual options for Lévy processes , 2002, Finance Stochastics.

[30]  B. M. Brown,et al.  High-Efficiency Estimation for the Positive Stable Laws , 1981 .

[31]  Yacine Aït-Sahalia,et al.  Disentangling diffusion from jumps , 2004 .

[32]  Ernst Eberlein,et al.  Term Structure Models Driven by General Lévy Processes , 1999 .

[33]  Marc Yor,et al.  Lévy processes in finance: a remedy to the non-stationarity of continuous martingales , 1998, Finance Stochastics.

[34]  V. M. Zolotarev,et al.  On Representation of Densities of Stable Laws by Special Functions , 1995 .

[35]  E. Fama,et al.  Some Properties of Symmetric Stable Distributions , 1968 .

[36]  S. Rachev,et al.  Stable Paretian Models in Finance , 2000 .

[37]  Svetlozar Rachev,et al.  Portfolio management with stable distributions , 2000, Math. Methods Oper. Res..

[38]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

[39]  ESTIMATING THE SKEWNESS IN DISCRETELY OBSERVED LÉVY PROCESSES , 2004, Econometric Theory.

[40]  Svetlozar T. Rachev,et al.  Stable modeling of value at risk , 2001 .

[41]  B. Mandelbrot The Variation of Certain Speculative Prices , 1963 .

[42]  Cecilia Mancini,et al.  Disentangling the jumps of the diffusion in a geometric jumping Brownian motion , 2001 .

[43]  Claudia Klüppelberg,et al.  Optimal portfolios when stock prices follow an exponential Lévy process , 2004, Finance Stochastics.

[44]  S. James Press,et al.  Estimation in Univariate and Multivariate Stable Distributions , 1972 .

[45]  P. Carr,et al.  What Type of Process Underlies Options? A Simple Robust Test , 2003 .

[46]  W. DuMouchel Estimating the Stable Index $\alpha$ in Order to Measure Tail Thickness: A Critique , 1983 .

[47]  N. Shephard,et al.  Power and bipower variation with stochastic volatility and jumps , 2003 .

[48]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[49]  Telling from Discrete Data Whether the Underlying Continuous-Time Model is a Diffusion , 2002 .

[50]  E. Fama The Behavior of Stock-Market Prices , 1965 .