On Some Applications of a Symbolic Representation of Non Centered Lévy Processes (original) (raw)

A new family of time-space harmonic polynomials with respect to Lévy processes

Annali di Matematica Pura ed Applicata, 2012

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to Lévy processes is given. The coefficients of these polynomials involve a formal expression of Lévy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson-Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of Lévy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to Lévy processes of very general form. We show the role played by cumulants of Lévy processes so that connections with boolean and free cumulants are also stated.

Time–space harmonic polynomials relative to a Lévy process

Bernoulli, 2008

In this work, we give a closed form and a recurrence relation for a family of time-space harmonic polynomials relative to a Lévy process. We also state the relationship with the Kailath-Segall (orthogonal) polynomials associated to the process.

On the orthogonal polynomials associated with a Lévy process

The Annals of Probability, 2008

Let X = {Xt, t ≥ 0} be a càdlàg Lévy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath-Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P1(x1), P2(x1, x2), . . . that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials p σ n (x) with respect to the measure σ 2 δ0(dx) + x 2 ν(dx), where σ 2 is the variance of the Gaussian part of X and ν its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Lévy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Lévy processes such that the associated polynomials Pn(x1, . . . , xn) depend on a fixed number of variables are characterized. Also, we give a sequence of Lévy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.

Multivariate Bernoulli and Euler polynomials via Lévy processes

Applied Mathematics Letters, 2012

By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate Lévy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate polynomials is provided.

Explicit identities for Lévy processes associated to symmetric stable processes

Bernoulli, 2011

In this paper we introduce a new class of Lévy processes which we call hypergeometricstable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.

PR ] 8 M ay 2 01 8 Lévy processes with respect to the index Whittaker convolution

2018

The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of Lévy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of Lévy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to Lévy’s characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.

Lévy processes with respect to the Whittaker convolution

Transactions of the American Mathematical Society, 2021

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures. In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution.

The extended hypergeometric class of Lévy processes

Journal of Applied Probability, 2014

We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.

A Stroock formula for a certain class of Lévy processes and applications to finance

Journal of Applied Mathematics and Stochastic Analysis, 2005

We find a Stroock formula in the setting of generalized chaos expansion introduced by Nualart and Schoutens for a certain class of Lévy processes, using a Malliavin-type derivative based on the chaotic approach. As applications, we get the chaotic decomposition of the local time of a simple Lévy process as well as the chaotic expansion of the price of a financial asset and of the price of a European call option. We also study the behavior of the tracking error in the discrete delta neutral hedging under both the equivalent martingale measure and the historical probability.

A ug 2 01 9 Lévy walk revisited : Hermite polynomial expansion approach

2019

Pengbo Xu, Weihua Deng, and Trifce Sandev School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia and Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia