Related papers: Approximation in law of locally $\alpha$-stable L\…
A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in $\mathbb{R}^d$ and whose generator behaves, locally, like an $\alpha$-stable L\'evy process, but the index $\alpha$ and all other characteristics may depend on the…
This paper is concerned with adaptive kernel estimation of the L\'evy density N(x) for bounded-variation pure-jump L\'evy processes. The sample path is observed at n discrete instants in the "high frequency" context (\Delta = \Delta(n)…
We consider a recurrent Markov process which is an It\^o semi-martingale. The L\'evy kernel describes the law of its jumps. Based on observations X(0),X({\Delta}),...,X(n{\Delta}), we construct an estimator for the L\'evy kernel's density.…
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…
We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to L\'evy-type processes…
Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of…
L\'evy stable (jump-type) processes are examples of intrinsically nonlocal random motions. This property becomes a serious obstacle if one attempts to model conditions under which a particular L\'evy process may be subject to physically…
We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary…
In this paper, we study the nonparametric estimation of the density $f_\Delta$ of an increment of a L\'evy process $X$ based on $n$ observations with a sampling rate $\Delta$. The class of L\'evy processes considered is broad, including…
We construct a Hunt process that can be described as an isotropic $\alpha$-stable L\'evy process reflected from the complement of a bounded open Lipschitz set. In fact, we introduce a new analytic method for concatenating Markov processes.…
We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a…
This paper is concerned with nonparametric estimation of the L\'evy density of a pure jump L\'evy process. The sample path is observed at $n$ discrete instants with fixed sampling interval. We construct a collection of estimators obtained…
We investigate an $L_{q}(L_{p})$-regularity ($1<p,q<\infty$) theory for space-time nonlocal equations of the type $\partial^{\alpha}_{t}u = \mathcal{L}u +f$. Here, $\partial^{\alpha}_{t}$ is the Caputo fractional derivative of order…
In this paper we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a time and path-dependent…
By using lower bound conditions of the L\'evy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical)…
Given a sample from a discretely observed L\'evy process $X=(X_t)_{t\geq 0}$ of the finite jump activity, the problem of nonparametric estimation of the L\'evy density $\rho$ corresponding to the process $X$ is studied. An estimator of…
We consider the parametric estimation of the Ornstein-Uhlenbeck process driven by a non-Gaussian $\alpha$-stable L\'{e}vy process with the stable index $\alpha>1$ and possibly skewed jumps, based on a discrete-time sample over a fixed…
This paper considers the martingale problem for a class of weakly coupled L\'{e}vy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process…
Efficient estimation of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a…
For $n$ equidistant observations of a L\'evy process at time distance $\Delta_n$ we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal-Getoor index in a non- or semiparametric manner.…