A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

A particular queue, the general arrival, general service-time, infinite-server queue (GA/G/∞), is introduced and certain of its properties studied. Motivated by a life situation in which the interarrival times for service converge to 0, a different sort of regularity condition (involving a tail property of random measures) is imposed on the arrival process to prove various limit theorems. There are similarities to heavy-traffic theory.

First it is shown that any generalized inverse of the infinitesimal generator of an irreducible Markov chain can be used to compute the exact stationary distribution and all the expected first-passage times of the chain. In the special case of a single-server queue this allows all computations to be done with upper-triangular matrices.

Next it is shown that the effect of a perturbation of the infinitesimal generator on the stationary distribution and expected first-passage times can also be computed using generalized inverses. These results extend and generalize Schweitzer's [9] original work using fundamental matrices. It is then shown that any perturbation can be broken up into a series of perturbations each involving a single state.

The Voronoi tessellation generated by a Gibbs point process is considered. Using the algebraic formalism of polymer expansion, the limit theorem and the large deviation principle for the number of Voronoi vertices are proved.

We consider the so-called frog model with random initial configurations. The dynamics of this model are described as follows. Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

We establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.

We consider generalisations of two classical stochastic scheduling models, namely the discounted branching bandit and the discounted multi-armed bandit, to the case where the collection of machines available for processing is itself a stochastic process. Under rather mild conditions on the machine availability process we obtain performance guarantees for a range of controls based on Gittins indices. Various forms of asymptotic optimality are established for index-based limit policies as the discount rate approaches 0.

A general way to look at kernel estimates of densities is to regard them as stochastic integrals with respect to a spatial point process. Under regularity conditions these behave asymptotically as if the point process were Poisson. However, this Poisson approximation may not work well if the data exhibits a lot of clustering. In this paper a more refined approximation to the characteristic functions of the integrals is developed. For clustered data, a ‘Gauss–Poisson’ approximation works better than the Poisson.

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

Motivated by the gene tree/species tree problem from statistical phylogenetics, we extend the class of Markov branching trees to a parametric family of distributions on fragmentation trees that satisfies a generalized Markov branching property. The main theorems establish important statistical properties of this model, specifically necessary and sufficient conditions under which a family of trees can be constructed consistently as sample size grows. We also consider the question of attaching random edge lengths to these trees.

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.

In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

In this paper a central limit theorem is proved for wave-functionals defined as the sums of wave amplitudes observed in sample paths of stationary continuously differentiable Gaussian processes. Examples illustrating this theory are given.

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.