Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes

In this paper we give easy to verify conditions for the strong consistency of the maximum likelihood estimator (MLE) in the case when data is sampled from a parametric family of selfdecomposable distributions. The difficulty arises from the fact that standard conditions for the consistency of the MLE are based on the pdf, which, for most selfdecomposable distributions, is not available in a closed form. Instead, our conditions are based on properties of the Lévy triplet (i.e. the Lévy measure, the Gaussian part, and the shift) of the distribution. Further, we extend out results to certain selfdecomposable stochastic processes, and, in particular, we give conditions in the case when the data is sampled from a Lévy or an Ornstein–Uhlenbeck process.     

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.     

Continuous-State Branching Processes in Lévy Random Environments

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than \(-1\). We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with blowup terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration.     

Branching Processes in a Lévy Random Environment

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.     

Growth of Lévy trees

We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure. T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries, EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779.     

Fractal properties of clusters generated by branching processes

An approach is developed to construction of random point distribution in 3-dimensional space based on the theory of branching processes. Correlation functions of all orders have been obtained in general form. The neutron and Lévy branching processes are numerically investigated. It is shown that fractal properties of the obtained distributions develop not only in the case of critical cascades but also in the case of subcritical, close to critical ones. In the last case, fractal properties appear in the limit region of distances. The difference between distributions generated by branching and nonbranching processes is discussed. Supported by the Russian State Committee on Higher Education (grant No. 95-0-3.1-23). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Jump tail dependence in Lévy copula models

This paper investigates the dependence of extreme jumps in multivariate Lévy processes. We introduce a measure called jump tail dependence, defined as the probability of observing a large jump in one component of a process given a concurrent large jump in another component. We show that this measure is determined by the Lévy copula alone and that it is independent of marginal Lévy processes. We derive a consistent nonparametric estimator for jump tail dependence and establish its asymptotic distribution. Regarding the economic relevance of the measure, a simulation study illustrates that jump tail dependence has a substantial impact on financial portfolio distributions and optimal portfolio weights.     

Stochastic equations and ergodicity for two-type continuous-state branching processes with immigration in Lévy random environments

This paper establishes a stochastic differential equation system with both positive and negative jumps and proves the existence and uniqueness of the strong solution and presents an equivalent condition for ergodicity of the solution. The strong solution is called two-type continuous-state branching processes with immigration in Lévy random environments. The model can be extended to any finite dimensional case.     

Selfdecomposable Fields

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein–Uhlenbeck processes.     

Estimates of densities for Lévy processes with lower intensity of large jumps

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.     

Local Subexponentiality and Self-decomposability

The class of exponential tilts of convolution equivalent distributions is determined. As a corollary, the local subexponentiality of one-sided infinitely divisible distributions is characterized. It is applied to the subexponentiality of the densities of a self-decomposable distribution and its Lévy measure. Bondesson’s conjecture on the density of the Lévy measure of a lognormal distribution is solved as an example. Results of Denisov et al. on the distributions of random sums are extended to the two-sided case. Finally, the local subexponentiality of the distribution of the supremum of a random walk is characterized.     

Fractional Lévy Processes as a Result of Compact Interval Integral Transformation

Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the com pact interval representation.

We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well.     

On a Class of Lévy Stochastic Networks

We consider a Lévy stochastic network as a regulated multidimensional Lévy process. The reflection direction is constant on each boundary of the positive orthant and the corresponding reflection matrix corresponds to a single-class network. We use the representation of the Lévy process and Itô's formula to arrive at some equations for the steady-state process; the latter is shown to exist, under natural stability conditions. We specialize first to the class of Lévy processes with non-negative jumps and then add the assumption of self-similarity. We show that the stationary distribution of the network corresponding the the latter process does not has product form (except in trivial cases). Finally, we derive asymptotic bounds for two-dimensional Lévy stochastic network.     

Hereditary tree growth and Lévy forests

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction. We only consider the metric structure of trees, and our framework is the space $\mathbb{T}$ of pointed isometry classes of locally compact rooted real trees equipped with the Gromov–Hausdorff distance. We discuss general tightness criteria in $\mathbb{T}$ and limit theorems for growing families of trees. We apply these results to Galton–Watson trees with exponentially distributed edge lengths. This class is preserved by hereditary reduction. Then we consider families of such Galton–Watson trees that are consistent under hereditary reduction and that we call growth processes. We prove that the associated families of offspring distributions are completely characterised by the branching mechanism of a continuous-state branching process. We also prove that such growth processes converge to Lévy forests. As a by-product of this convergence, we obtain a characterisation of the laws of Lévy forests in terms of leaf-length erasure and we obtain invariance principles for discrete Galton–Watson trees, including the super-critical cases.     

Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Lévy processes with periodic mean

We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process.     

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.     

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.