We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use comparison relationships between networks of this class with independent and identically distributed driving sequences and the GI/GI/1/1 queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times. The exponential rate of decay is given as a function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with examples arising in queueing theory and insurance risk.

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.

In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

We consider basic ergodicity properties of adaptive Markov chain Monte Carlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.

We present bounds on the decay parameter for absorbing birth–death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nåsell (2001).

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

We consider ageing properties of a general repair process. Under certain assumptions we prove that the expectation of an age at the beginning of the next cycle in this process is smaller than the initial age of the previous cycle. Using this reasoning, we show that the sequence of random ages at the start (end) of consecutive cycles is stochastically increasing and is converging to a limiting distribution. We discuss possible applications and interpretations of our results.

Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.