Speed of convergence to the quasi-stationary distribution for L\'evy input fluid queues

In this note we prove that the speed of convergence of the workload of a L\'evy-driven queue to the quasi-stationary distribution is of order $1/t$. We identify also the Laplace transform of the measure giving this speed and provide some examples

On future drawdowns of Levy processes

For a given Levy process X = ( X t ) t 2 R + and for xed s 2 R + [f1g and t 2 R + we analyse the future drawdown extremes that are de ned as follows: The path-functionals D t;s and D t;s are of interest in various areas of application, including nancial mathematics and queueing theory. In the case that X has a strictly positive mean, we nd the exact asymptotic decay as x ! 1 of the tail probabilities P ( D t < x ) and P ( D t < x ) of D t = lim s !1 D t;s and D t = lim s !1 D t;s both when the jumps satisfy the Cram er assumption and in a heavy-tailed case. Furthermore, in the case that the jumps of the L evy process X are of single sign and X is not subordinator, we identify the one-dimensional distributions in terms of the scale function of X . By way of example, we derive explicit results for the Black- Scholes-Samuelson model

On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk process in the presence of a penalty function

This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed.Comment: Published at http://dx.doi.org/10.1214/14-AAP1038 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A L\'evy input fluid queue with input and workload regulation

We consider a queuing model with the workload evolving between consecutive i.i.d.\ exponential timers $\{e_q^{(i)}\}_{i=1,2,...}$ according to a spectrally positive L\'evy process $Y_i(t)$ that is reflected at zero, and where the environment $i$ equals 0 or 1. When the exponential clock $e_q^{(i)}$ ends, the workload, as well as the L\'evy input process, are modified; this modification may depend on the current value of the workload, the maximum and the minimum workload observed during the previous cycle, and the environment $i$ of the L\'evy input process itself during the previous cycle. We analyse the steady-state workload distribution for this model. The main theme of the analysis is the systematic application of non-trivial functionals, derived within the framework of fluctuation theory of L\'evy processes, to workload and queuing models

Maxima over random time intervals for heavy-tailed compound renewal and Lévy processes

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a Lévy process, both with negative drift, over random time horizon τ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by τ independent of the processes. We link our results with random walk theory

Extremes of multitype branching random walks: Heaviest tail wins

We consider a branching random walk on a multi(Q)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index α, while the other types of particles have lighter tails than that of particles of type Q. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPPP) using the tools developed in \cite{bhattacharya:hazra:roy:2016}. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the nth generation

Extremes of multitype branching random walks: Heaviest tail wins

We consider a branching random walk on a multitype (with Q types of particles), supercritical Galton-Watson tree which satisfies the Kesten-Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index α, while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the nth generation

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

We consider the sample average of a centered random walk in Rd with regularly varying step size distribution. For the first exit time from a compact convex set A not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter