Lowest priority waiting time distribution in an accumulating priority Lévy queue

We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of an infinitesimal customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. To this end we introduce a new method that relies on the construction of a workload overtaking process and solving a first-passage problem using an appropriate stopping time.     

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.     

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].     

Product and moment formulas for iterated stochastic integrals (associated with Lévy processes)

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables us to consider Lévy processes, with both jumps and Gaussian part.     

On the refracted–reflected spectrally negative Lévy processes

We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.     

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.

In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

Weighted empirical processes in the nonparametric inference for Lévy processes

Given observations of a Lévy process, we provide nonparametric estimators of its Lévy tail and study the asymptotic properties of the corresponding weighted empirical processes. Within a special class of weight functions, we give necessary and sufficient conditions that ensure strong consistency and asymptotic normality of the weighted empirical processes, provided that complete information on the jumps is available. To cope with infinite activity processes, we depart from this assumption and analyze the weighted empirical processes of a sampling scheme where small jumps are neglected. We establish a bootstrap principle and provide a simulation study for some prominent Lévy processes.     

On Lévy-driven vacation models with correlated busy periods and service interruptions

This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Lévy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server’s preceding busy period. Regarding the former point: the Lévy process active during the busy period is assumed to have no negative jumps, whereas the Lévy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537–555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.     

Asymptotic analysis of Lévy-driven tandem queues

We analyze tail asymptotics of a two-node tandem queue with spectrally-positive Lévy input. A first focus lies in the tail probabilities of the type ?(Q ₁>α x,Q ₂>(1?α)x), for α∈(0,1) and x large, and Q _i denoting the steady-state workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.     

High order schemes for the tempered fractional diffusion equations

Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering Lévy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the α-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Grünwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.     

Multifractal scenarios for products of geometric Lévy-based stationary models

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We construct the multifractal, such as log-gamma, log-tempered stable, or log-normal tempered stable scenarios.     

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.      

Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.      

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.     

Predicting the Time at Which a Lévy Process Attains Its Ultimate Supremum

We consider the problem of finding a stopping time that minimises the L ¹-distance to θ, the time at which a Lévy process attains its ultimate supremum. This problem was studied in Du Toit and Peskir (Proc. Math. Control Theory Finance, pp. 95–112, 2008) for a Brownian motion with drift and a finite time horizon. We consider a general Lévy process and an infinite time horizon (only compound Poisson processes are excluded. Furthermore due to the infinite horizon the problem is interesting only when the Lévy process drifts to ?∞). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If θ has infinite mean there exists no stopping time with a finite L ¹-distance to θ, whereas if θ has finite mean it is either optimal to stop immediately or to stop when the process reflected in its supremum exceeds a positive level, depending on whether the median of the law of the ultimate supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the Lévy process has no positive jumps.     

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.     

On the location of the maximum of a process: Lévy,Gaussian and Random field cases

ABSTRACT

We show how the techniques presented in Pimentel [On the location of the maximum of a continuous stochastic process, J. Appl. Prob. 51 (2014), pp. 152–161] can be extended to a variety of non-continuous processes and random fields. For the Gaussian case, we prove new covariance formulae between the maximum and the maximizer of the process. As examples, we prove uniqueness of the location of the maximum for spectrally positive Lévy processes, Ornstein–Uhlenbeck process, fractional Brownian Motion and the Brownian sheet among other processes.     

A risk model driven by Lévy processes

We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd.     

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.