Area Littlewood–Paley Functions Associated with Hermite and Laguerre Operators

In this paper we study L ^p —boundedness properties for area Littlewood–Paley functions associated with heat semigroups for Hermite and Laguerre operators.     

Lévy–Khintchine Representations of the Weighted Geometric Mean and the Logarithmic Mean

In the paper, the authors introduce the concept “degree of complete Bernstein function”, establish Lévy–Khintchine representations of the weighted geometric mean and the logarithmic mean by Cauchy integral formula, and obtain that the weighted geometric mean and the logarithmic mean are complete Bernstein functions of degree 0.     

On the Lévy–Khintchine representation of q-Bessel negative definite functions

We study q-Bessel negative definite functions and use their properties to give a proof of Lévy–Khintchine representation of q-infinitely divisible probability measures.     

Hypercyclic and Chaotic Convolution Associated with the Jacobi–Dunkl Operator

In this paper, we study the Jacobi–Dunkl convolution operators on some distribution spaces. We characterize the Jacobi–Dunkl convolution operators as those ones that commute with the Jacobi–Dunkl translations and with the Jacobi–Dunkl operators. Also we prove that the Jacobi–Dunkl convolution operators are hypercyclic and chaotic on the spaces under consideration and we give a universality property for the generalized heat equation associated with them.     

Stochastic Processes Associated with a Sum of the Lévy Laplacians

Abstract

In this paper, we give a stochastic expression of a semigroup generated by a sum of the Lévy Laplacians acting on a class of S-transforms of white noise distributions in terms of an infinite sequence of independent Brownian motions.     

Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

Journal of Theoretical Probability - We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha...     

Derivative Formula and Harnack Inequality for SDEs Driven by Lévy Processes

By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.     

Derivative Formula and Coupling Property for Linear SDEs Driven by Lévy Processes

Acta Mathematicae Applicatae Sinica, English Series - In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding...     

Multiple integration with respect to Poisson and Lévy processes

Summary Necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form ...fdX ₁...dX_d, where X ₁, ..., X _d are components of a positive or symmetric pure jump type Lévy process in _d. Conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity, or to be tight. All arguments proceed via reduction to the special case of Poisson integrals.Dedicated to Klaus Krickeberg on the occasion of his 60th birthdaySupported by NSF grant DMS-8703804Supported by NSF grant DMS-8713103     

Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Singular SDEs with critical non-local and non-symmetric Lévy type generator

In this work, by using Levi’s parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the pathwise uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.     

A Lévy input fluid queue with input and workload regulation

We consider a controlled queuing model with Lévy input. The controls take place at random times. They involve the current workload and the input processes and may also depend on whether the workload process has reached certain critical values since the last control epoch. We propose a solution strategy for deriving the steady-state distribution of this model which is based on recent advances in the fluctuation theory of spectrally one-sided Lévy process. We provide illustrative examples involving a clearing model, an inventory model, and a model for the TCP protocol.     

On the Lévy–Raikov–Marcinkiewicz theorem

Let μ be a finite nonnegative Borel measure. The classical Lévy–Raikov–Marcinkiewicz theorem states that if its Fourier transform $\text{μ}\text{?}$ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then $\text{μ}\text{?}$ admits analytic continuation into the strip $\text{{t:}\text{0<}\text{I}\text{t<R}}$. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Pricing currency derivatives with Markov-modulated Lévy dynamics

Using a Lévy process we generalize formulas in Bo et al. (2010) for the Esscher transform parameters for the log-normal distribution which ensure that the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.