laurent Loosveldt | Searcharxiv

arXiv2026

Modified weighted power variations of the Hermite process and applications to integrated volatility

We study the asymptotic behaviour of modified weighted power variations of the Hermite process of arbitrary order. By selecting suitable "good" increments and exploiting their decomposition into dominant independent components, we establish a central limit theorem for weighted $p$-variations using tools from Stein-Malliavin calculus. Our results extend previous works on modified quadratic and wavelet-based variations to general powers and to weighted settings, with explicit bounds in Wasserstein distance. We further apply these limit theorems to construct asymptotically Gaussian estimators of integrated volatility in Hermite-driven models, thereby extending fBm-based methods to non-Gaussian settings. The last part of our work contains numerical simulations which illustrate the practical performance of the proposed estimators.

math.ST↗

arXiv2025

Numerical simulation of Generalized Hermite Processes

Hermite processes are paradigmatic examples of stochastic processes which can belong to any Wiener chaos of an arbitrary order; the wellknown fractional Brownian motion belonging to the Gaussian first order Wiener chaos and the Rosenblatt process belonging to the non-Gaussian second order Wiener chaos are two particular cases of them. Except these two particular cases no simulation method for sample paths of Hermite processes is available so far. The goal of our article is to introduce a new method which potentially allows to simulate sample paths of any Hermite process and even those of any generalized Hermite process. Our starting point is the representation for the latter process as random wavelet-typeseries, obtained in our very recent paper [3]. We construct from it a "concrete" sequence of piecewise linear continuous random functions which almost surely approximate sample paths of this process for the uniform norm on any compact interval, and we provide an almost sure estimate of the approximation error. Then, for the Rosenblatt process and more importantly for the third order Hermite process, we propose algorithms allowing to implement this sequence and we illustrate them by several simulations. Python routines implementing these synthesis procedures are available upon request.

math.PR↗