The purpose of this paper is to analyse the real-time trading of electricity. We address a model for an auction-like trading which captures key features of real-world electricity markets. Our main result establishes that, under certain conditions, the expected total payment for electricity is independent of the particular auction type. This result is analogous to the revenue-equivalence theorem known for classical auctions and could contribute to an improved understanding of different electricity market designs and their comparison.

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ 0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λ n , n = 1,2,…, as well as an integral over the continuous spectrum above μ 2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ 0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

A method is provided for numerical evaluation, with any given accuracy, of the probability that at least p% of the genetic material from an individual's chromosomal segment survives to the next generation. Relevant MAPLE® V codes, for automated implementation of such evaluation, are also provided. The genomic continuum model, with Haldane's model for the crossover process, is assumed.

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X t − may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X t − means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.

Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.

A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.

In this paper we extend some recent results on the comparison of multivariate risk vectors with respect to supermodular and related orderings. We introduce a dependence notion called the ‘weakly conditional increasing in sequence order’ that allows us to conclude that ‘more dependent’ vectors in this ordering are also comparable with respect to the supermodular ordering. At the same time, this ordering allows us to compare two risks with respect to the directionally convex order if the marginals increase convexly. We further state comparison criteria with respect to the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally, we discuss Fréchet bounds with respect to Δ-monotone functions when multivariate marginals are given. It turns out that, in the case of multivariate marginals, comonotone vectors no longer yield necessarily the largest risks but, in some cases, may even be vectors which minimize risk.

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

This paper considers a class of continuous-time long-range-dependent Gaussian processes. The corresponding spectral density is assumed to have a general and flexible form, which covers some important and special cases. For example, the spectral density of a continuous-time fractional stochastic differential equation is included. A modelling procedure is then established through estimating the parameters involved in the spectral density by using an extended continuous-time version of the Gauss–Whittle objective function. The resulting estimates are shown to be strongly consistent and asymptotically normal. An application of the modelling procedure to the identification and modelling of a fractional stochastic volatility is discussed in some detail.

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.

This paper presents a single-server loss queueing system where customers arrive according to a Poisson process. Upon arrival, the customer presents itself to a gatekeeper who has to decide whether to admit the customer into the system without knowing the busy–idle status of the server. There is a cost if the gatekeeper blocks a customer, and a larger cost if an admitted customer finds the server busy and therefore has to leave the system. The goal of the gatekeeper is to minimize the total expected discounted cost on an infinite time horizon. In the case of an exponential service distribution, we show that a threshold-type policy—block for a time period following each admission and then admit the next customer—is optimal. For general service distributions, we show that a threshold-type policy need not be optimal; we then present a sufficient condition for the existence of an optimal threshold-type policy.

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.