We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let Pn and Mn denote the respective numbers of plus balls and minus balls drawn by time n and define Z0 = 0, Zn = Pn - Mn, 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.

We give a general construction of sequential games among multiple players, as well as a construction of the composition of sequential games. We obtain new properties of the optimal class of win-by-k games, including closure under composition and independence between the winner of the game and the number of points played. We obtain new results on the asymptotic efficiency of the n-point, win-by-k games.

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

We consider the sum Sd of record values in a sequence of independent random variables that are uniformly distributed on 1,…,d. This sum can be interpreted as the total amount of time spent in record lifetimes in the standard renewal theoretic setup. We investigate the distributional limit of Sd and some related quantities as d→∞. Some explicit values are given for d=6, a case that can be interpreted as a simple game of chance.

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

The paper yields retrieval formulae of the directional distribution of a stationary k-flat process in ℝd if its rose of intersections with all r-flats is known. Cases k = d −1, 1 ≤ r ≤ d - 1 for arbitrary d and d = 4, k = 2, r = 2 are considered. Some generalizations to manifold processes in ℝd are made. The proofs use the methods of harmonic analysis on higher Grassmannians (spherical harmonics, integral transforms).

With the aim of providing greater flexibility in developing and applying shot noise models, this paper studies shot noise on cluster point processes with both pointwise and cluster marks. For example, in financial modelling, responses to events in the financial market may occur in clusters, with random amplitudes including a ‘cluster component’ reflecting a degree of commonness among responses within a cluster. For such shot noise models, general formulae for the characteristic functional are developed and specialized to the case of Neyman-Scott clustering with cluster marks. For several general forms of response function, long range dependence of the corresponding equilibrium shot noise models is investigated. It is shown, for example, that long range dependence holds when the ‘structure component’ of the response function decays slowly enough, or when the response function has a finite random duration with a heavy tailed distribution.

We solve the following three optimal stopping problems for different kinds of options, based on the Black-Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l, where x is the current stock price.

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

A new statistical method for estimating the orientation distribution of fibres in a fibre process is suggested where the process is observed in the form of a degraded digital greyscale image. The method is based on line transect sampling of the image in a few fixed directions. A well-known method based on stereology is available if the intersections between the transects and fibres can be counted. We extend this to the case where, instead of the intersection points, only scaled variograms of grey levels along the transects are observed. The nonlinear estimation equations for a parametric orientation distribution as well as a numerical algorithm are given. The method is illustrated by a real-world example and simulated examples where the elliptic orientation distribution is applied. In its simplicity, the new approach is intended for industrial on-line estimation of fibre orientation in disordered fibrous materials.

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

Intrinsic volumes are key functionals in convex geometry and have also appeared in several stochastic settings. Here we relate them to questions of regularity in Gaussian processes with regard to Itô–Nisio oscillation and metrization of GB/GC indexing sets. Various bounds and estimates are presented, and questions for further investigation are suggested. From alternate points of view, much of the discussion can be interpreted in terms of (i) random sets and (ii) properties of (deterministic) infinite-dimensional convex bodies.

This paper analyzes optimal single and multiple stopping rules for a class of correlated random walks that provides an elementary model for processes exhibiting momentum or directional reinforcement behavior. Explicit descriptions of optimal stopping rules are given in several interesting special cases with and without transaction costs. Numerical examples are presented comparing optimal strategies to simpler buy and hold strategies.

We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.

Let X = (X(t):t ≥ 0) be a Lévy process and X∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X∊(1)). In simulation, X - X∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.