Estimation for Lévy processes from high frequency data within a long time interval

Statistics and Modeling for Complex Data
COMTE Fabienne

This talk is semi-parametric in the sense that we simultaneously estimate a function and two parameters. More precisely, we study nonparametric estimation of the Lévy density for Lévy processes, first without then with Brownian component. We consider $2n$ (resp. $3n$) discrete time observations with step $\Delta$. The asymptotic framework is: $n$ tends to infinity, $\Delta=\Delta_n$ tends to zero while $n\Delta_n$ tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global $L^2$-risk.

More precisely, we consider $(L_t, t \ge 0)$ a real-valued Lévy process, i.e. a process with stationary independent increments and càdlàg sample paths with associated Lévy measure admitting a density $n(x)$. We assume that the Lévy density satisfies $\int_{R} x^2 n(x) dx < \infty.$ For statistical purposes, this assumption, which was proposed in Neumann and Reiss (2009), has several useful consequences. First, for all $t$, $EL_t^2<+\infty$ and as $\int_{R}(e^{iux} -1-iux)n(x)dx$ is well defined, we get: $$\label{fc} \psi_{t}(u)= E(\exp{i u L_t})=\exp{t(iu b-\frac 12 u^2\sigma^2 +\int_{R}(e^{iux} -1-iux) n(x)dx)},(1)$$ where $b=E(L_1)$. Formula (1) is the starting point of the nonparametric part of the estimation strategy and provides, by derivating twice (if $\sigma=0$) or three times (in the general case) an estimator of $h(x)=x^2n(x)$ (if $\sigma=0$) or $p(x)=x^3n(x)$ (in the general case). These estimators are defined by Fourier inversion of type $$\hat p_m(x)= (2\pi)^{-1} \int_{-\pi m}^{\pi m} e^{-iux}\hat p^*(x)dx,$$ and the cutoff parameter $m$ is chosen by a penalization device to obtain the best possible squared bias-variance compromise. Risk bounds are provided for the integrated mean square risk of the adaptive estimators. We discuss rates of convergence and give examples of processes fitting in our framework, for which we can compute the explicit rates of convergence.

Estimators of the drift and of the variance of the Gaussian component are also studied, based for $\sigma^2$ on power variations in the spirit of Aït-Sahalia and Jacod (2007). Simulation results for such processes are also given, for functions $g(x)=xn(x)$, $h(x)=x^2n(x)$ and $p(x)=x^3n(x)$ and also for the parameter estimates of $b$ and $\sigma^2$.


Adaptive nonparametric estimation; High frequency data; L ́evy processes; Projection estimators; Power variation.


Comte, F. and Genon-Catalot, V. (2011). Estimation for L ́evy processes from high frequency data within a long time interval. Ann. Statist., to appear.
Neumann, M. and Reiss, M. (2009). Nonparametric estimation for L ́evy processes from low-frequency observations. Bernoulli 15, 223-248.
Aït-Sahalia Y. and Jacod J. (2007). Volatility estimators for discretely sampled L ́evy processes. The Annals of Statistics 35, 355-392.