Related papers: L\'evy Processes, Generalized Moments and Uniform …
The classical notion of L\'evy process is generalized to one that takes as its values probabilities on a first order model equipped with a commutative semigroup. This is achieved by applying a convolution product on definable probabilities…
It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions…
We study critera for a pair $ (\{ X_n \} $, $ \{ Y_n \}) $ of approximating processes which guarantee closeness of moments by generalizing known results for the special case that $ Y_n = Y $ for all $n$ and $ X_n $ converges to $Y$ in…
Starting from the potential theoretic definition of the local times of a Markov process - when these exist - we obtain a Tanaka formula for the local times of symmetric L\'{e}vy processes. The most interesting case is that of the symmetric…
Distributional properties -including Laplace transforms- of integrals of Markov processes received a lot of attention in the literature. In this paper, we complete existing results in several ways. First, we provide the analytical solution…
We generalise multivariate subordination of L\'evy processes as introduced by Barndorff-Nielsen, Pedersen, and Sato to Hilbert space valued L\'evy processes. The processes are explicitly characterised and conditions for integrability and…
We obtain a representation of an inhomogeneous Levy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Because the stochastic continuity is not assumed, our result generalizes the…
It is possible to construct L\'evy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two…
This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calder\'on integrability condition, the temperateness condition and the uniform counting estimate. It is shown that…
In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric L\'evy processes. More precisely, we prove that a one-dimensional symmetric L\'evy process with the L\'evy measure $\nu(dy)=f(y)dy$ or…
The one dimensional distribution of a L\'{e}vy process is not known in general even though its characteristic function is given by the famous L\'{e}vy-Khinchine theorem. This article gives an exact series representation for the one…
In this article we consider L\'evy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample…
We generalize a fundamental theorem on positive matrix semigroups stating that each component is either strictly positive for all times or identically zero ("L\'evy's Theorem"). Our proof of this fact that does not require the matrices to…
The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R…
In this work, we consider moments of exponential functionals of L\'{e}vy processes on a deterministic horizon. We derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential…
Long memory processes driven by L\'evy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here,…
We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily…
We provide asymptotic results and develop high frequency statistical procedures for time-changed L\'evy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with…
Let $X=\{X(t),t\in R_+\}$ be a real-valued symmetric L\'{e}vy process with continuous local times $\{L^x_t,(t,x)\in R_+\times R\}$ and characteristic function $Ee^{i\lambda X(t)}=e^{-t\psi(\lambda)}$. Let…
For an arbitrary L\'evy process $X$ which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of $X$…