Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

Abstract

A Lévy model combines a Brownian motion with drift and a purejump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling scheme. In that case, the jumps are latent variables whose statistical properties can in principle be assessed when the frequency of observations increase to infinity. We propose nonparametric estimators for the Lévy density following Grenander's method of sieves. The associated problem of selecting a suitable approximating sieve is subsequently investigated using regular piece-wise polynomials as sieves and assuming standard smoothness conditions on the Lévy density. By sampling the process at a high enough frequency relative to the time horizon T , we show that it is feasible to choose the dimension of the sieve so that the rate of convergence of the risk of estimation off the origin is the best possible from a minimax point of view, and even if the estimation were based on the whole sample path of the process. The sampling frequency necessary to attain the optimal minimax rate is explicitly identified. The proposed method is illustrated by simulation experiments in the case of variance Gamma processes.

Nonparametric estimation of time-changed Lévy models under high-frequency data

“…In the case of a true Lévy process, Figueroa-López [13] proved this result assuming that ϕ is bounded, ν continuous, and such that ϕ(x) = o(|x|) as x → 0. The random clock case is more challenging, as in this caseβ n is not the sum of independent random variables.…”

“…It is natural to devise statistical methodologies based on highfrequency observations since this type of data will contain more relevant information about the jumps of the process and, hence, about the Lévy density s. Such a high-frequency-based statistical approach has played a central role in the recent literature on nonparametric estimation for Lévy processes (see, e.g. [10], [13], [20], [24], and [25]). For instance, under discrete 1164 J. E. FIGUEROA-LÓPEZ observations of a pure Lévy process X at times π : 0 = t 0 < · · · < t n = T , Woerner [24], and also independently Figueroa-López [10], proposed the estimatorŝ…”

“…In [10], the proposed estimators were used to devise nonparametric estimatorsŝ for s via Grenander's method of sieves. The problem of model selection was analyzed further in [13] and [15], where it was proved that sieve estimatorss T can match the rate of convergence of the minimax risk of estimatorsŝ. Concretely, it turns out that lim sup…”

“…The optimal rate of the estimators T is attained by appropriately choosing the dimension of the sieve and the sampling frequency as a function of T and the smoothness of the class of functions . In [11], the sieve estimators of [13] were also used to build confidence intervals and confidence bands for the Lévy density s.…”

Adaptive Estimation for Lévy Processes

“…on a compact set separated from the origin (see e.g. [31]). Settingn(x) =ĝ(x)/x, we have the obvious inequality Analogous inequalities hold forn(x) =l(x)/x 2 orn(x) =p(x)/x 3 .…”