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.     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.     

On the regularity of the local times of a class of symmetric lévy processes

We provevia Dynkin's isomorphism theorem, that spatial trajectories of local times of a class of symmetric Lévy processes, with regularly varying Lévy exponent ψ at infinity, belong to a class of Besov spaces. Our result generalizes the case of symmetric stable Lévy processes treated in [5]     

On the infimum attained by a reflected Lévy process

This paper considers a Lévy-driven queue (i.e., a Lévy process reflected at 0), and focuses on the distribution of M(t), that is, the minimal value attained in an interval of length t (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided Lévy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of ℙ(M(T _u )>u) (for different classes of functions T _u and u large); here we have to distinguish between heavy-tailed and light-tailed scenarios.     

A Functional LIL for <Emphasis Type="Italic">d</Emphasis>-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes

We establish a functional LIL for the maximal process M(t) :=sup _0≤s≤t ‖X(s)‖ of an ℝ ^d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X. Supported in part by NSF Grant DMS 02-05034.     

Asymptotic theory for estimating the parameters of a Lévy process

Summary We consider consistency and asymptotic normality of maximum likelihood estimators (MLE) for parameters of a Lévy process of the discontinuous type. The MLE are based on a single realization of the process on a given interval [0,t]. Depending on properties of the Lévy measure we either consider the MLE corresponding to jumps of size greater than ε and, keepingt fixed, we let ε tend to 0, or we consider the MLE corresponding to the complete information of the realization of the process on [0,t] and lett tend to ∞. The results of this paper improve in both generality and rigor previous asymptotic estimation results for such processes.     

Quelques nouvelles identités de fluctuation pour les processus de Lévy

Let τ and H be the ladder time and ladder height processes of a Lévy process X. We give an identity in law between (τ,H) and (X,H^*), H^* being the right continuous inverse of the process H. The later allows us to get a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (X_t- inf_s≤ X_s >- 0). In the stable case, some explicit calculations are provided.     

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.     

Pricing and Hedging of Lookback Options in Hyper-exponential Jump Diffusion Models

ABSTRACT

In this article, we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Lévy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A transform approach to calculate prices and Greeks of barrier options driven by a class of Lévy processes. Quntitative Finance, 10(6), 629–644.) were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalized hyper-exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments.     

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.     

Itô's formula for C 1,λ -functions of a càdlàg process and related calculus

 This article develops a framework of stochastic calculus with respect to a càdlàg finite quadratic variation process. We apply it to the study of a generalization of a semimartingale driven SDE studied by Kurtz, Pardoux and Protter [KPP]. We prove an It?'s formula for functions f(X) of a semimartingale with jumps when f has weak smoothness properties. Examples of X for which this formula is valid are time reversible semimartingales and solutions of [KPP] equations driven by Lévy processes, provided the sum of the absolute values of the jumps, raised to the power 1 + λ, is a.s. finite, where λ takes values between 0 and 1. Received: 1 March 1999 / Revised version: 15 April 2001 / Published online: 11 December 2001     

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.     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Approximation of the occupation measure of Lévy processes

Consider a real-valued Lévy process with non-zero Gaussian component and jumps with locally finite variation. We obtain an invariance principle theorem for the speed of approximation of its occupation measure by means of functionals defined on regularizations of the paths. This theorem is a first extension to processes with jumps of previous results for semimartingales with continuous paths. To cite this article: E. Mordecki, M. Wschebor, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Statistical specification of jumps under semiparametric semimartingale models

We consider a semimartingale with jumps that are driven by a finite activity Lévy process. Suppose that the Lévy measure is completely unknown, and that the jump component has a Markovian structure depending on unknown parameters. This paper concentrates on estimating the parameters from continuous observations under the nonparametric setting on the Lévy measure. The estimating function is proposed by way of nonparametric approach for some regression functions. In the end, we can specify jumps of the underlying Lévy process and estimate some Lévy characteristics jointly.      

Fragmentation associated with Lévy processes using snake

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.      

M-Estimation for Discretely Observed Ergodic Diffusion Processes with Infinitely Many Jumps

We study parametric inference for multidimensional stochastic differential equations with jumps from some discrete observations. We consider a case where the structure of jumps is mainly controlled by a random measure which is generated by a Lévy process with a Lévy measure f_θ(z)dz, and we admit the case ∫ f_θ(z)dz = ∞ in which infinitely many small jumps occur even in any finite time intervals. We propose an estimating function under this complicated situation and show the consistency and the asymptotic normality. Although the estimators in this paper are not completely efficient, the method can be applied to comparatively wide class of stochastic differential equations, and it is easy to compute the estimating equations. Therefore, it may be useful in applications. We also present some simulation results for some simple models. Final version 25 December 2004     

Regularity for semigroups of Ornstein–Uhlenbeck processes

Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L ^p (dx) → L ^q (dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein–Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein–Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known.