Related papers: L\'{e}vy driven CARMA generalized processes and st…
A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the…
This paper concerns the numerical approximation for the invariant distribution of Markovian switching L\'evy-driven stochastic differential equations. By combining the tamed-adaptive Euler-Maruyama scheme with the Multi-level Monte Carlo…
Uncertainties are abundant in complex systems. Mathematical models for these systems thus contain random effects or noises. The models are often in the form of stochastic differential equations, with some parameters to be determined by…
The article studies non-Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and Gaussian Markov random fields. The focus is…
By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and L\'evy processes with…
This paper considers a continuous time analogue of the classical autoregressive moving average processes, L\'evy-driven CARMA processes. First we describe limiting properties of the periodogram by means of the so-called truncated Fourier…
We study the extremal behavior of a stochastic integral driven by a multivariate L\'{e}vy process that is regularly varying with index $\alpha>0$. For predictable integrands with a finite $(\alpha+\delta)$-moment, for some $\delta>0$, we…
This brief manuscript provides an introduction to L\'evy processes and their applications in finance as the random process that drives asset models. Characteristic functions and random variable generators of popular L\'evy processes are…
We describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by L\'evy noise. We first discuss in details the finite dimensional case with a linear, resp. non…
The purpose of this note is to prove that the flatness of an invariant manifold for a semilinear stochastic partial differential equation driven by L\'{e}vy processes is at least equal to the number of driving sources with small jumps. We…
We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In…
We present a new code that significantly extends CRPropa's capabilities to model the ensemble averaged transport of charged cosmic rays in arbitrary turbulent magnetic fields. The software is based on solving a set of stochastic…
Estimation methods for the L\'{e}vy density of a L\'{e}vy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good…
In this article we consider parametric Bayesian inference for stochastic differential equations (SDE) driven by a pure-jump stable Levy process, which is observed at high frequency. In most cases of practical interest, the likelihood…
We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a L\'evy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the L\'evy measure of…
The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic…
Additive processes are obtained from L\'{e}vy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can define…
We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…
Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose…
In this paper approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional…