Donsker's Delta Function of Lévy Process

Let X={X(t):tR} be a Lévy process and a non-decreasing, right continuous, bounded function with (–)=0 (((1+u ²)/u ²)d(u) is the Lévy measure). In this paper we define the Donsker delta function (X(t)–a), t>0 and aR, as a generalized Lévy functional under the condition that (0)–(0–)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Itô formula for F(X(t)) when has jumps at 0 and 1.     

Reflected BSDE Driven by a Lévy Process

In this paper, we study the reflected solution of one-dimensional backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove the existence and uniqueness of the solution using a penalization method combined with Snell envelope theory.      

Functionals of a Lévy Process on Canonical and Generic Probability Spaces

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable \(Y\) by a functional \(F\) mapping from the Skorohod space of càdlàg functions to \(\mathbb {R}\), such that \(Y=F(X)\) where \(X\) denotes the Lévy process. We also present a chain-rule-type application for random variables of the form \(f(\omega ,Y(\omega ))\). An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Solé et al.) associated to a Lévy process with triplet \((\gamma ,\sigma ,\nu )\) to an arbitrary probability space \((\varOmega ,\mathcal {F},\mathbb {P})\) which carries a Lévy process with the same triplet.     

Asymptotic Behavior of Eigenvalues of Variance-Covariance Matrix of a High-Dimensional Heavy-Tailed Lévy Process

Methodology and Computing in Applied Probability - In this paper, we study the limiting behavior of eigenvalues of the variance-covariance matrix of a random sample from a multivariate subordinator...     

The Coding Complexity of Lévy Processes

We introduce a new coding scheme for general real-valued Lévy processes and control its performance with respect to L _p [0,1]-norm distortion under different complexity constraints. We also establish lower bounds that prove the optimality of our coding scheme in many cases.      

Approximations of a Complex Brownian Motion by Processes Constructed from a Lévy Process

Mediterranean Journal of Mathematics - In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments....     

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     

Optimal Harvesting for a Logistic Population Dynamics Driven by a Lévy Process

The optimal harvesting problem for a stochastic logistic jump-diffusion process is studied in this paper. Two kinds of environmental noises are considered in the model. One is called white noise which is described by a standard Brownian motion, and the other is called jumping noise which is described by a Lévy process. For three types of yield functions (time averaging yield, expected yield and sustainable yield), the optimal harvesting efforts, the corresponding maximum yields and the steady states of population mean under optimal harvesting strategy are respectively given. A new equivalent relationship among these three different objective functions is showed by the ergodic method. This method provides a new approach to the optimal harvesting problem. Results in this paper show that environmental noises have important effect on the optimal harvesting problem.     

The Lévy–Khinchin representation of a class of stable signed measures

We study properties of symmetric stable measures with index of stability α ∈ (2, 4) ∪ (4, 6). For such signed measures, we construct a natural analog of the Lévy-Khinchin representation. We show that, in some special sense, these measures are limit measures for sums of independent random variables. Bibliography: 6 titles. Translated from Zapiski Nauchnykh. Seminarov POMI, Vol. 361, 2008, pp. 145–166.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

The Lévy Swap Market Model

Models driven by Lévy processes are attractive since they allow for better statistical fitting than classical diffusion models. The dynamics of the forward swap rate process is derived in a semimartingale setting and a Lévy swap market model is introduced. In order to guarantee positive rates, the swap rates are modelled as ordinary exponentials. The model starts with the most distant rate, which is driven by a non‐homogeneous Lévy process. Via backward induction the remaining swap rates are constructed such that they become martingales under the corresponding forward swap measures. Finally it is shown how swaptions can be priced using bilateral Laplace transforms.     

The Most Visited Sites of Certain Lévy Processes

Let X be a symmetric Lévy process with

On a Stochastic Wave Equation Driven by a Non-Gaussian Lévy Process

This paper investigates a damped stochastic wave equation driven by a non-Gaussian Lévy noise. The weak solution is proved to exist and be unique. Moreover we show the existence of a unique invariant measure associated with the transition semigroup under mild conditions.     

The law of the iterated logarithm for local time of a Lévy process

Summary Let {X _t } be a one-dimensional Lévy process with local timeL(t, x) andL ^*(t)=sup{L(t, x): x }. Under an assumption which is more general than being a symmetric stable process with index >1, we obtain a LIL forL*(t). Also with an additional condition of symmetry, a LIL for range is proved.This research is supported by a grant from Korea Science and Engineering Foundations     

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.     

Nonsingular transformations of a class of tempered Lévy processes

Let ξ(t), t ∈ [0, 1], be a tempered stable process. By {ie4447-01} we denote the law of ξ in the Skorokhod space {ie4447-02}. For the measure {ie4447-03}, we construct the group of nonsingular transformations of {ie4447-04}. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 38–53.     

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.