Statistical inference for time-changed L{e}vy processes via composite characteristic function estimation

arXiv:1003.0275v3 [stat.ME] 30 Jan 2012 The Annals of Statistics 2011, Vol. 39, No. 4, 2205–2242 DOI: 10.1214/11-AOS901 c ⃝Institute of Mathematical Statistics, 2011 STATISTICAL INFERENCE FOR TIME-CHANGED L´EVY PROCESSES VIA COMPOSITE CHARACTERISTIC FUNCTION ESTIMATION By Denis Belomestny Duisburg-Essen University In this article, the problem of semi-parametric inference on the parameters of a multidimensional L´evy process Lt with independent components based on the low-frequency observations of the corre- sponding time-changed L´evy process LT (t), where T is a nonnega- tive, nondecreasing real-valued process independent of Lt, is studied. We show that this problem is closely related to the problem of com- posite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the L´evy density of Lt and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed L´evy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) L´evy processes. 1. Introduction. The problem of nonparametric statistical inference for jump processes or more generally for semimartingale models has long history and goes back to the works of Rubin and Tucker (1959) and Basawa and Brockwell (1982). In the past decade, one has witnessed the revival of interest in this topic which is mainly related to a wide availability of financial and economical time series data and new types of statistical issues that have not been addressed before. There are two major strands of recent literature dealing with statistical inference for semimartingale models. The first type of literature considers the so-called high-frequency setup, where the asymptotic properties of the corresponding estimates are studied under the assumption Received December 2010; revised April 2011. 1Supported in part by SFB 649 “Economic Risk.” AMS 2000 subject classifications. Primary 62F10; secondary 62J12, 62F25, 62H12. Key words and phrases. Time-changed L´evy processes, dependence, pointwise and uni- form rates of convergence, composite function estimation. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2011, Vol. 39, No. 4, 2205–2242. This reprint differs from the original in pagination and typographic detail. 1 2 D. BELOMESTNY that the frequency of observations tends to infinity. In the second strand of literature, the frequency of observations is assumed to be fixed (the so-called low-frequency setup) and the asymptotic analysis is done under the premiss that the observational horizon tends to infinity. It is clear that none of the above asymptotic hypothesis can be perfectly realized on real data and they can only serve as a convenient approximation, as in practice the frequency of observations and the horizon are always finite. The present paper studies the problem of statistical inference for a class of semimartingale models in low-frequency setup. Let X = (Xt)t≥0 be a stochastic process valued in Rd and let T = (T (s))s≥0 be a nonnegative, nondecreasing stochastic process not necessarily indepen- dent of X with T (0) = 0. A time-changed process Y = (Ys)s≥0 is then defined as Ys = XT (s). The process T is usually referred to as time change. Even in the case of the one-dimensional Brownian motion X, the class of time- changed processes XT is very large and basically coincides with the class of all semimartingales [see, e.g., Monroe (1978)]. In fact, the construction in Monroe (1978) is not direct, meaning that the problem of specification of different models with the specific properties remains an important issue. For example, the base process X can be assumed to possess some independence property (e.g., X may have independent components), whereas a nonlinear time change can induce deviations from the independence. Along this line, the time change can be used to model dependence for stochastic processes. In this work, we restrict our attention to the case of time-changed L´evy pro- cesses, that is, the case where X = L is a multivariate L´evy process and T is an independent of L time change. Time-changed L´evy processes are one step further in increasing the complexity of models in order to incorporate the so-called stylized features of the financial time series, like volatility clus- tering [for more details, see Carr et al. (2003)]. This type of processes in the case of the one-dimensional Brownian motion was first studied by Bochner (1949). Clark (1973) introduced Bochner’s time-changed Brownian motion into financial economics: he used it to relate future price returns of cotton to the variations in volume during different trading periods. Recently, a nu

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