Related papers: On Piterbarg Max-discretisation Theorem for Multiv…
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
The fluctuation-dissipation theorem is a fundamental result in statistical physics that establishes a connection between the response of a system subject to a perturbation and the fluctuations associated with observables in equilibrium.…
The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in…
We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We…
We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models…
Consider the max-stable process $\eta(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - \kappa(t)}$, $t\in\mathbb{R}^d$, where $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ on…
We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first…
For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes which can be obtained as limits, as $n\to\infty$, of the process $a_n(M_n-b_n)$, where…
We study the asymptotic behavior of the supremum $M_t$ of the support of a supercritical super-Brownian motion. In our recent paper (Stoch. Proc. Appl. 137 (2021), 1-34), we showed that, under some conditions, $M_t-m(t)$ converges in…
We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on graph (lattice) with different…
Let $\bigl\{X_k\bigr\}_{k \in \mathbb{Z}} \in \mathbb{L}^2(\mathcal{T})$ be a stationary process with associated lag operators ${\boldsymbol{\cal C}}_h$. Uniform asymptotic expansions of the corresponding empirical eigenvalues and…
In this paper we describe a triple correspondence between graph limits, information theory and group theory. We put forward a new graph limit concept called log-convergence that is closely connected to dense graph limits but its main…
With any max-stable random process $\eta$ on $\mathcal{X}=\mathbb{Z}^d$ or $\mathbb{R}^d$, we associate a random tessellation of the parameter space $\mathcal{X}$. The construction relies on the Poisson point process representation of the…
Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $\sigma^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For…
This paper studies small-time behavior at the supremum of a diffusion process. For a solution to the SDE $\mathrm{d} X_t=\mu(X_t)\mathrm{d} t+\sigma(X_t)\mathrm{d} W_t$ (where $W$ is a standard Brownian motion) we consider…
Higher criticism is a large-scale testing procedure that can attain the optimal detection boundary for sparse and faint signals. However, there has been a lack of knowledge in most existing works about its asymptotic distribution for more…
An important result in classical stochastic thermodynamics is the work fluctuation--dissipation relation (FDR), which states that the dissipated work done along a slow process is proportional to the resulting work fluctuations. Here we show…
We study the limit of the joint distribution of a multidimensional Generalized Tempered Stable (GTS) process and its quadratic covariation process when the stable index tends to two. Under a proper scaling, the GTS processes converges to a…
In a classical problem for the stopping of a diffusion process $(X_t)_{t \geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$, maximisation…
The present work investigates two properties of level crossings of a stationary Gaussian process $X(t)$ with autocorrelation function $R_X(\tau)$. We show firstly that if $R_X(\tau)$ admits finite second and fourth derivatives at the…