Related papers: H\"older regularity for stochastic processes with …
We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of density functions…
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range…
It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff…
The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs…
Stochastic estimators are fundamental to large-scale optimization, where population quantities must be inferred from noisy oracle observations. Although influential methods such as momentum, SPIDER, STORM, and PAGE have been highly…
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any…
The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete…
We continue the investigation of the spectral theory and exponential asymptotics of Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, characterizing distinct subclasses…
In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal $\ell^p$-regularity in…
We present a systematic derivation of the algorithms required for computing the gradient and the action of the Hessian of an arbitrary misfit function for large-scale parameter estimation problems involving linear time-dependent PDEs with…
Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate…
We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of…
We consider a wide range of regularized stochastic minimization problems with two regularization terms, one of which is composed with a linear function. This optimization model abstracts a number of important applications in artificial…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
Parameter estimation for a parabolic linear stochastic partial differential equation in one space dimension is studied observing the solution field on a discrete grid in a fixed bounded domain. Considering an infill asymptotic regime in…
Let $\Phi:\R\rightarrow\R$ be an arbitrary continuously differentiable deterministic function such that $|\Phi|+|\Phi'|$ is bounded by a polynomial. In this article we consider the class of stochastic volatility models in which…
In traditional work on numerical schemes for solving stochastic differential equations (SDEs), it is usually assumed that the coefficients are globally Lipschitz. This assumption has been used to establish a powerful analysis of the…
Piecewise-deterministic Markov processes (PDMPs) offer a powerful stochastic modeling framework that combines deterministic trajectories with random perturbations at random times. Estimating their local characteristics (particularly the…
The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space.…
This paper is devoted to the estimation of the common marginal density function of weakly dependent processes. The accuracy of estimation is measured using pointwise risks. We propose a datadriven procedure using kernel rules. The bandwidth…