arXiv subjects

Taher Jalal

Publications and source records attributed to Taher Jalal.

3 recordsLinked to original sources

arXiv2024

Stable and tempered stable distributions and processes: an overview toward trajectory simulation

Stable distributions are a celebrated class of probability laws used in various fields. The $α$-stable process, and its exponentially tempered counterpart, the Classical Tempered Stable (CTS) process, are also prominent examples of Lévy processes. Simulating these processes is critical for many applications, yet it remains computationally challenging, due to their infinite jump activity. This survey provides an overview of the key properties of these objects offering a roadmap for practitioners. The first part is a review of the stability property, sampling algorithms are provided along with numerical illustrations. Then CTS processes are presented, with the Baeumer-Meerschaert algorithm for increment simulation, and a computational analysis is provided with numerical illustrations across different time scales.

math.PR↗

arXiv2024

Nonparametric density estimation for the small jumps of Lévy processes

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high frequency observations. In a low frequency setting, assuming the Lévy density associated with the jumps larger than $\varepsilon\in (0,1]$ in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated $L_2$ loss, up to a logarithmic factor. In a high frequency setting, we remove the assumption on the knowledge of the Lévy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behaviour of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we illustrate numerically the performances of our procedures.

math.ST↗

arXiv2024

Adaptive minimax estimation for discretely observed Lévy processes

In this paper, we study the nonparametric estimation of the density $f_Δ$ of an increment of a Lévy process $X$ based on $n$ observations with a sampling rate $Δ$. The class of Lévy processes considered is broad, including both processes with a Gaussian component and pure jump processes. A key focus is on processes where $f_Δ$ is smooth for all $Δ$. We introduce a spectral estimator of $f_Δ$ and derive both upper and lower bounds, showing that the estimator is minimax optimal in both low- and high-frequency regimes. Our results differ from existing work by offering weaker, easily verifiable assumptions and providing non-asymptotic results that explicitly depend on $Δ$. In low-frequency settings, we recover parametric convergence rates, while in high-frequency settings, we identify two regimes based on whether the Gaussian or jump components dominate. The rates of convergence are closely tied to the jump activity, with continuity between the Gaussian case and more general jump processes. Additionally, we propose a fully data-driven estimator with proven simplicity and rapid implementation, supported by numerical experiments.

math.ST↗