We introduce a machine-learning framework named statistics-informed neural network
(SINN) for learning stochastic dynamics from data. This new architecture was theoretically …

Consider the following stochastic differential equation (SDE) d X t= b (t, X t−) d t+ d L t, X 0=
x, driven by a d-dimensional Lévy process (L t) t≥ 0. We establish conditions on the Lévy …

Due to the increasing power of modern computers, it has become ever more popular to use
sophisticated models to describe the irregular behaviour of real life phenomena. Nowadays …

We address estimation of parametric coefficients of a pure-jump Lévy driven univariate
stochastic differential equation (SDE) model, which is observed at high frequency over a …

These lecture notes are an extended version of my lectures on Lévy and Lévy-type
processes given at the Second Barcelona Summer School on Stochastic Analysis organized …

In this paper we study the domain of the generator of stable processes, stable-like processes
and more general pseudo-and integro-differential operators which naturally arise both in …

In this paper, we study the Besov regularity of a general d-dimensional Lévy white noise.
More precisely, we describe new sample paths properties of a given noise in terms of …

A maximal inequality is an inequality which involves the (absolute) supremum sup s⩽ t| X s|
or the running maximum sup s⩽ t X s of a stochastic process (X t) t⩾ 0. We discuss maximal …

We present an existence result for Lévy‐type processes which requires only weak regularity
assumptions on the symbol with respect to the space variable x. Applications range from …

We characterize the local smoothness and the asymptotic growth rate of the Lévy white
noise. We do so by characterizing the weighted Besov spaces in which it is located. We …