This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.

We introduce two general classes of reflected autoregressive processes, INGAR+ and GAR+. Here, INGAR+ can be seen as the counterpart of INAR(1) with general thinning and reflection being imposed to keep the process non-negative; GAR+ relates to AR(1) in an analogous manner. The two processes INGAR+ and GAR+ are shown to be connected via a duality relation. We proceed by presenting a detailed analysis of the time-dependent and stationary behavior of the INGAR+ process, and then exploit the duality relation to obtain the time-dependent and stationary behavior of the GAR+ process.

The extinction probability q of a supercritical simple branching process is well known to be less than unity. Intuitively, it is apparent that when the offspring mean is close to one, so, usually, will q be. This notion is made rigorous, and simple bounds are given for q in terms of the second and third factorial moments, which are applicable when the offspring mean is close to unity. A comparison is made of various upper bounds for q. The note contains some numerical examples.

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

We investigate the conditions on a hedger, who overestimates the (time- and level-dependent) volatility, to superreplicate a convex claim on several underlying assets. It is shown that the classic Black-Scholes model is the only model, within a large class, for which overestimation of the volatility yields the desired superreplication property. This is in contrast to the one-dimensional case, in which it is known that overestimation of the volatility with any time- and level-dependent model guarantees superreplication of convex claims.

We consider a stochastic SIR (susceptible → infective → removed) epidemic on a random graph with specified degree distribution, constructed using the configuration model, and investigate the ‘acquaintance vaccination’ method for targeting individuals of high degree for vaccination. Branching process approximations are developed which yield a post-vaccination threshold parameter, and the asymptotic (large population) probability and final size of a major outbreak. We find that introducing an imperfect vaccine response into the present model for acquaintance vaccination leads to sibling dependence in the approximating branching processes, which may then require infinite type spaces for their analysis and are generally not amenable to numerical calculation. Thus, we propose and analyse an alternative model for acquaintance vaccination, which avoids these difficulties. The theory is illustrated by a brief numerical study, which suggests that the two models for acquaintance vaccination yield quantitatively very similar disease properties.

When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable.

In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Consider an optimization problem for a company with the following parameters: a constant liability payment rate (δ), an average return (μ) and a risk (σ) proportional to the size of the business unit, and an internal competition factor (α) between different units. The goal is to maximize the expected present value of the total dividend distributions, via controls (Ut, Zt), where Ut is the size of the business unit and Zt is the total dividend payoff up to time t. We formulate this as a stochastic control problem for a diffusion process Xt and derive an explicit solution by solving the corresponding Hamilton-Jacobi-Bellman equation. The resulting optimal control policy involves a mixture of a nonlinear control for Ut and a singular control for Zt. The optimal strategies are different for the cases δ < 0 and δ = 0. When δ > 0, it is optimal to play bold: the initial optimal investment size should be proportional to the debt rate δ. Under this optimal rule, however, the probability of bankruptcy in finite time is 1. When δ = 0, i.e. when the company is free of debt, the probability of going broke in finite time reduces to 0. Moreover, when δ = 0, the value function is singular at X0 = 0. Our analytical result shows considerable consistency with daily business practices. For instance, it shows that ‘too many people is counter-productive’. In fact, the maximal optimal size of the business unit should be inversely proportional to α. This eliminates the redundant and simplistic technical assumption of a known uniform upper bound on the size of the firm.

The asymptotic behavior of the maximum likelihood estimator of a parameter in the drift term of a stationary ergodic diffusion process is studied under conditions in which the true drift function and true noise function do not coincide with those specified by the parametric model.

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.

For the original Moran dam with independent and identically distributed inputs a representation of the stationary distribution is given which readily provides a geometric rate of convergence to this distribution. For the integer-valued case the stationary distribution can be expressed in terms of simple boundary crossing probabilities for the underlying random walk.

Much of the theory of epidemics (see Bailey, 1957) has been concerned with models for their behaviour in closed populations. In such models the epidemic ultimately dies out, and interest has been concentrated on, for example, the ultimate size of the epidemic and its duration in time. In practice a population is rarely completely closed, and for many diseases an endemic model rather than an epidemic model is appropriate. To create models for endemic diseases it is necessary to introduce both new persons susceptible to the disease into the population and new sources of infection. For the so-called general stochastic epidemic, Ridler-Rowe (1967) has obtained certain limiting properties of the population where immigration of both susceptibles and infectives into the population takes place, but much work remains to be done to obtain, for example, the general equilibrium behaviour of this model.

Classical epidemic models have invariably proved to be mathematically intractable. By considering the distribution of the number of infectives in a simple epidemic process as a convolution of exponential waiting times, the solution to the classical model is obtained easily giving more insight into the underlying structure. The idea can be extended to other simple epidemic models.

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.