arXiv subjects

Valentin Garino

Publications and source records attributed to Valentin Garino.

6 recordsLinked to original sources

arXiv2025

Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs

In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function $\sigma$ in the stochastic partial differential equation $Lu = \sigma(u)\dot{W}$, where $L$ is a parabolic second order differential operator and $\dot{W}$ is a suitable Gaussian noise. We prove consistency and derive an upper bound for the error in the $L^p$ norm, in terms of discretization and smoothening parameters $h$ and $\varepsilon$. We illustrate the applicability of the approach and the role of the parameters with several interesting numerical examples.

math.ST↗

arXiv2023

On the Impact of Approximation Errors on Extreme Quantile Estimation with Applications to Functional Data Analysis

We study the effect of approximation errors in assessing the extreme behavior of heavy-tailed random objects. We give conditions for the approximation error such that the standard asymptotic results hold for the classical Hill estimator and the corresponding extreme quantile estimator. As an application, we consider the effect of discretization errors in the computation of the $L^p$-norms related to functional data. We approximate the norms both with Riemann sums and with Monte Carlo integration. We quantify connections between the number of observed functions, the number of discretization points, and the regularity of the underlying functions. In addition, we derive a new concentration inequality for order statistics. This, to the best of our knowledge, is the first Chernoff-type concentration inequality for order statistics presented in the literature that provides an explicit rate at which the ratio between order statistics and tail quantile function converges to one. In our application, the bound is used to provide concentration inequalities measuring the distance between the Hill estimator based on approximated norms and the one based on the true ones.

math.ST↗

arXiv2023

Discretisation error for stochastic integrals with respect to the fractional Brownian motion with discontinuous integrands and local times

We consider equidistant Riemann approximations of stochastic integrals $\int_0^T f(B^H_s)dB^H_s$ with respect to the fractional Brownian motion with $H>\frac12$, where $f$ is an arbitrary function of locally bounded variation, hence possibly possessing discontinuities. We prove that properly normalised approximation error converge in the $L^2$-topology to a functional of the local time, and we provide rate of convergence for this approximation. As such, our results complements some recent advances on the topic as well as provides new methods for simulation of local times.

math.PR↗

arXiv2023

Total variation bound for Hadwiger's functional using Stein's method

Let $K$ be a convex body in $\mathbb{R}^d$. Let $X_K$ be a $d$-dimensional random vector distributed according to the Hadwiger-Wills density $\mu_K$ associated with $K$, defined as $\mu_K(x)=ce^{-\pi {\rm dist}^2(x,K)}$, $x\in \mathbb{R}^d$. Finally, let the information content $H_K$ be defined as $H_K={\rm dist}^2(X_K,K)$. The goal of this paper is to study the fluctuations of $H_K$ around its expectation as the dimension $d$ go to infinity. Relying on Stein's method and Brascamp-Lieb inequality, we compute an explicit bound for the total variation distance between $H_K$ and its Gaussian counterpart.

math.PR↗

arXiv2020

Limit theorems for integral functionals of Hermite-driven processes

Consider a moving average process $X$ of the form $X(t)=\int_{-\infty}^t x(t-u)dZ_u$, $t\geq 0$, where $Z$ is a (non Gaussian) Hermite process of order $q\geq 2$ and $x:\mathbb{R}_+\to\mathbb{R}$ is sufficiently integrable. This paper investigates the fluctuations, as $T\to\infty$, of integral functionals of the form $t\mapsto \int_0^{Tt }P(X(s))ds$, in the case where $P$ is any given polynomial function. It extends a study initiated in Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.

math.PR↗

arXiv2020

Asymptotic error distribution for the Riemann approximation of integrals driven by fractional Brownian motion

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-It\^o integrals

math.PR↗