We set up a model for electronic foreign-exchange markets, suggesting subordinators to represent sellers' and buyers' offers. Its analysis naturally leads to the study of level passage events. The classical level passage event concerns the joint law of the time, height, and jump size observed when a real-valued stochastic process first exceeds a given level h. We provide an up-to-date treatment in the case when this process is a subordinator, and extend these results to a multivariate setting.

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

A simplified proof of Thorp and Walden's fundamental theorem of card counting is presented, and a corresponding central limit theorem is established. Results are applied to the casino game of trente et quarante, which was studied by Poisson and De Morgan.

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ 0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

Throw n points sequentially and at random onto a unit circle and append a clockwise arc (or rod) of length s to each such point. The resulting random set (the free gas of rods) is a union of a random number of clusters with random sizes modelling a free deposition process on a one-dimensional substrate. A variant of this model is investigated in order to take into account the role of the disorder, θ > 0; this involves Dirichlet(θ) distributions. For such free deposition processes with disorder θ, we shall be interested in the occurrence times and probabilities, as n grows, of two specific types of configurations: those avoiding overlapping rods (the hard-rod gas) and those for which the largest gap is smaller than the rod length s (the packing gas). Special attention is paid to the thermodynamic limit when ns = ρ for some finite density ρ of points. The occurrence of parking configurations, those for which hard-rod and packing constraints are both fulfilled, is then studied. Finally, some aspects of these problems are investigated in the low-disorder limit θ ↓ 0 as n ↑ ∞ while nθ = γ > 0. Here, Poisson-Dirichlet(γ) partitions play some role.

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

It is well known that the Kelly system of proportional betting, which maximizes the long-term geometric rate of growth of the gambler's fortune, minimizes the expected time required to reach a specified goal. Less well known is the fact that it maximizes the median of the gambler's fortune. This was pointed out by the author in a 1988 paper, but only under asymptotic assumptions that might cause one to question its applicability. Here we show that the result is true more generally, and argue that this is a desirable property of the Kelly system.

We consider a failure-prone system operating in continuous time. Condition monitoring is conducted at discrete time epochs. The state of the system is assumed to evolve as a continuous-time Markov process with a finite state space. The observation process with continuous-range values is stochastically related to the state process, which, except for the failure state, is unobservable. Combining the failure information and the condition monitoring information, we derive a general recursive filter, and, as special cases, we obtain recursive formulae for the state estimation and other quantities of interest. Updated parameter estimates are obtained using the expectation-maximization (EM) algorithm. Some practical prediction problems are discussed and finally an illustrative example is given using a real dataset.

Let X n ,…,X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(G α), where G α(x) = [1 - exp(-x α)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

We continue the study of the asymptotic behaviour of a random walk when it exits from a symmetric region of the form {(x, n): |x| ≤ rn b } as r → ∞ which was begun in Part I of this work. In contrast to that paper, we are interested in the case where the probability of exiting at the upper boundary tends to 1. In this scenario we treat the case where the power b lies in the interval [0, 1), and we establish necessary and sufficient conditions for the overshoot to be relatively stable in probability (except for the case ), and for the pth moment of the overshoot to be O(r q ) as r → ∞.

Let (S n )n≥0 be a correlated random walk on the integers, let M 0 ≥ S 0 be an arbitrary integer, and let M n = max{M 0, S 1,…, S n }. An optimal stopping rule is derived for the sequence M n - nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that M n - S n ≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.

Suppose that μ is the branching measure on the boundary of a supercritical Galton–Watson tree with offspring distribution N such that E[N log N] < ∞ and P{N = 1} > 0. We determine the multifractal spectrum of μ using a method different from that proposed by Shieh and Taylor, which is flawed.

We introduce a new model for the infection of one or more subjects by a single agent, and calculate the probability of infection after a fixed length of time. We model the agent and subjects as random walkers on a complete graph of N sites, jumping with equal rates from site to site. When one of the walkers is at the same site as the agent for a length of time τ, we assume that the infection probability is given by an exponential law with parameter γ, i.e. q(τ) = 1 - e-γτ . We introduce the boundary condition that all walkers return to their initial site (‘home’) at the end of a fixed period T. We also assume that the incubation period is longer than T, so that there is no immediate propagation of the infection. In this model, we find that for short periods T, i.e. such that γ T ≪ 1 and T ≪ 1, the infection probability is remarkably small and behaves like T 3. On the other hand, for large T, the probability tends to 1 (as might be expected) exponentially. However, the dominant exponential rate is given approximately by 2γ/[(2+γ)N] and is therefore small for large N.

We study a family of locally self-similar stochastic processes Y = {Y(t)}t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

In this paper, we show that the mean comparison theorem, which is valid for Brownian motion, cannot be extended to Poisson processes. A counterexample in the Poisson case for which the mean comparison theorem does not hold is provided.

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.