Harmonic analysis of additive Lévy processes

Let X ₁, . . . ,X _N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets ${F\subseteq{{\bf R}^d}}$ , the Minkowski sum ${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$ , of the range ${\mathfrak{X}({{\bf R}^{N}_{+}})}$ of ${\mathfrak{X}}$ with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.     

Lebesgue measure of the range of additive Lévy processes

We prove a theorem on the Lebesgue measure of the range of additive Lévy Processes and then use this theorem to remove Condition (1.3) of Theorem 1.5 of Khoshnevisan et al. (Ann Probab 31:1097–1141, 2003).     

The first hitting time of the integers by symmetric Lévy processes

For one-dimensional Brownian motion, the exit time from an interval has finite exponential moments and its probability density is expanded in exponential terms. In this note we establish its counterpart for certain symmetric Lévy processes. Applying the theory of Pick functions, we study properties of the Laplace transform of the first hitting time of the integer lattice as a meromorphic function in detail. Its density is expanded in exponential terms and the poles and the zeros of a Pick function play a crucial role.Intermediate results concerning finite exponential moments are also obtained for a class of nonsymmetric Lévy processes.     

Abrupt Lévy processes

Among Lévy processes with unbounded variation, we distinguish the abrupt ones, which are characterised by infinitely sharp extrema. Stable processes with parameter α>1 and creeping Lévy processes are abrupt. We give a characterisation of abrupt processes and study their Dini derivatives at all points of their trajectories.     

On the regularity of the local times of a class of symmetric lévy processes

We provevia Dynkin's isomorphism theorem, that spatial trajectories of local times of a class of symmetric Lévy processes, with regularly varying Lévy exponent ψ at infinity, belong to a class of Besov spaces. Our result generalizes the case of symmetric stable Lévy processes treated in [5]     

Structural properties of reflected Lévy processes

This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V _t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.     

The multifractal nature of Lévy processes

We show that the sample paths of most Lévy processes are multifractal functions and we determine their spectrum of singularities. Received: 21 February 1997 / Revised version: 27 July 1998     

Approximations for the distributions of bounded variation Lévy processes

We propose a feasible method for approximating the marginal distributions and densities of a bounded variation Lévy process using polynomial expansions. We provide a fast recursive formula for approximating the coefficients of the expansions and estimating the order of the approximation error. Our expansions are shown to be the exact counterpart of successive approximations of the Lévy process by compound Poisson processes previously proposed by, for instance, Barndorff-Nielsen and Hubalek (2008) [Barndorff-Nielsen, O.E., Hubalek, F., 2008. Probability measures, Lévy measures, and analyticity in time. Bernoulli 3 (14), 764–790] and others, and hence, give an answer to an open problem raised therein.     

On the limiting behaviour of Lévy processes at zero

We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to and as where X is a Lévy Process and T _r the time it first leaves an interval of radius r     

On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes1

We investigate a sufficient condition for pathwise uniqueness property for 1D stochastic differential equation driven by symmetric α-stable Lévy process, where α ∈ (1, 2).     

Optimal stopping of Markov switching Lévy processes

We consider a finite time horizon optimal stopping of a regime-switching Lévy process. We prove that the value function of the optimal stopping problem can be characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequalities.     

Fluctuations of Omega-killed spectrally negative Lévy processes

In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases $\omega \left(x\right)=q$ and $\omega \left(x\right)=q{\mathbf{1}}_{(a,b)}\left(x\right)$, we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate $\omega \left(x\right)$ when the Lévy surplus process is at level $x<0$. Finally, we apply these results to obtain some exit identities for spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.     

On hypoellipticity of generators of Lévy processes

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., \(\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu\) for all admissible u. In particular, we assume that \(\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})\). We also show that this condition is necessary provided that \(\mathop {\mathrm {supp}}\mu\) is compact.     

Asymptotical properties of distributions of isotropic Lévy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1.The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi \left(\lambda \right)?\frac{\lambda }{2}{\psi }^{\prime }\left(\lambda \right)$ varies regularly, as $\frac{t\psi {\left(r^{?1}\right)}^2}{\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)}\to 0$ the tail probability $–(\left|X_t\right|\ge r)$ is asymptotically equal to a constant times $t(\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)).$     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

Measure preserving transformations of jump Lévy processes

Let x(t),t ? [ 0,1 ] \xi (t),t \in \left[ {0,1} \right] , be a jump Lévy process. By P_x {\mathcal{P}_\xi } we denote the law of in the Skorokhod space \mathbbD {\mathbb{D}} [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of P_x {\mathcal{P}_\xi } -preserving transformations of the space \mathbbD {\mathbb{D}} [0, 1]. Bibliography: 10 titles.     

Weighted empirical processes in the nonparametric inference for Lévy processes

Given observations of a Lévy process, we provide nonparametric estimators of its Lévy tail and study the asymptotic properties of the corresponding weighted empirical processes. Within a special class of weight functions, we give necessary and sufficient conditions that ensure strong consistency and asymptotic normality of the weighted empirical processes, provided that complete information on the jumps is available. To cope with infinite activity processes, we depart from this assumption and analyze the weighted empirical processes of a sampling scheme where small jumps are neglected. We establish a bootstrap principle and provide a simulation study for some prominent Lévy processes.     

The Hausdorff Dimension of Operator Semistable Lévy Processes

Let X={X(t)} _t≥0 be an operator semistable Lévy process in ? ^d with exponent E, where E is an invertible linear operator on ? ^d and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao (Stoch. Process. Appl. 115, 55–75, 2005) for the special case of an operator stable (selfsimilar) Lévy process, for an arbitrary Borel set B??₊ we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.