The Segal-Bargmann transform for Lévy white noise functionals associated with non-integrable Lévy processes

By using a method of truncation, we derive the closed form of the Segal-Bargmann transform of Lévy white noise functionals associated with a Lévy process with the Lévy spectrum without the moment condition. Besides, a sufficient and necessary condition to the existence of Lévy stochastic integrals is obtained.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

A Decomposition Theorem for Lévy Processes on Local Fields

The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.     

Occupation times of spectrally negative Lévy processes with applications

In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative Lévy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative Lévy process and its Laplace exponent. Applications to insurance risk models are also presented.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

Lévy-Driven Carma Processes

Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.     

A Stochastic Process Generated by the Lévy Laplacian

In this paper we give a stochastic process generated by the Lévy Laplacian in the white noise analysis with a characterization of the Laplacian.     

Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes

In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier-Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts.     

On a Class of Non-Translation Invariant Feller Semigroups on Lie Groups

We consider a class of Feller semigroups on Lie groups which fail to commute with left translation due to the existence of a cocycle h which is identically one for Lévy processes. Under certain conditions, we are able to show that the infinitesimal generator of such a semigroup has the Lévy–Khintchine–Hunt form but with variable characteristics, thus we obtain an extension of classical work in Euclidean space by Courrège.     

Symmetrization of Lévy processes and applications

It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs.     

Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

Occupation times of intervals until first passage times for spectrally negative Lévy processes

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative Lévy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented.     

Mortality modelling with Lévy processes

This paper addresses the modelling of human mortality by the aid of doubly stochastic processes with an intensity driven by a positive Lévy process. We focus on intensities having a mean reverting stochastic component. Furthermore, driving Lévy processes are pure jump processes belonging to the class of α-stable subordinators. In this setting, expressions of survival probabilities are inferred, the pricing is discussed and numerical applications to actuarial valuations are proposed.     

Adaptive nonparametric estimation for Lévy processes observed at low frequency

This article deals with adaptive nonparametric estimation for Lévy processes observed at low frequency. For general linear functionals of the Lévy measure, we construct kernel estimators, provide upper risk bounds and derive rates of convergence under regularity assumptions.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

Lévy processes and homogeneous killing transformation

In this paper we introduce homogeneous multiplicative functionals of Lévy processes and investigate their bivariate Revuz measures. We also prove that the sector condition will be inherited by sub-Lévy processes through the killing transformation.This research is supported in part by Zhejiang University Science Foundation.     

Ratio Limit Theorems for Random Walks and Lévy Processes

In this paper, we study quasi-symmetric random walks and Lévy processes, a property first introduced by C.J. Stone, discuss the -invariant Radon measures for random walks and Lévy processes, and formulate some nice ratio limit theorems which are closely related to -invariant Radon measures. Mathematics Subject Classifications (2000) 60G51, 60G50.Research supported in part by NSFC 10271109.     

Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support

It is shown that a Lévy white noise measure Λ always exists as a Borel measure on the dual K^′ of the space K of C^∞ functions on R with compact support. Then a characterization theorem that ensures that the measurable support of Λ is contained in S^′ is proved. In the course of the proofs, a representation of the Lévy process as a function on K^′ is obtained and stochastic Lévy integrals are studied.