The Palm version of a stationary random subset of a d-dimensional grid is contructed using the two-step change-of-origin and change-of-measure method. An elementary proof is given of the fact that the Palm version is characterized by point-stationarity (distributional invariance under bijective shifts of the origin from a point of the set to another point of the set).

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem.

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution Pn is weighted by giving the path S=(S 0,…,S n ) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.

We consider the Voronoi tessellation based on a stationary Poisson process N in ℝd . We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

The random surface measure of a stationary Boolean model with grains from the convex ring is considered. A sufficient condition and a necessary condition for the existence of the density of the second-order moment measure of are given and a representation of this density is derived. As applications, the surface pair correlation functions of a Boolean model with spheres and a Boolean model with randomly oriented right circular cylinders in ℝ3 are determined.

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show interesting size effects.

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar measures. Our results are useful in understanding the multifractal nature of various heterogeneous jump processes.

In this paper we investigate sufficient conditions that ensure the optimality of threshold strategies for optimal stopping problems with finite or perpetual maturities. Our result is based on a local-time argument that enables us to give an alternative proof of the smooth-fit principle. Moreover, we present a class of optimal stopping problems for which the propagation of convexity fails.

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ ½ ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let S n =ξ1+⋯+ξn and M n =maxk≤n S k . Let τ=min{n≥1: S n ≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

This paper presents a study of the intertemporal propagation of distributional properties of phenotypes in general polygenic multisex inheritance models with sex- and time-dependent heritabilities. It further analyzes the implications of these models under heavy-tailedness of traits' initial distributions. Our results suggest the optimality of a flexible asexual/binary mating system. Switching between asexual and binary inheritance mechanisms allows the population effectively to achieve a fast suppression of negative traits and a fast dispersion of positive traits, regardless of the distributional properties of the phenotypes in the initial period.

The volume fraction of the intact grains of the dead leaves model with spherical grains of equal size is 2−d in d dimensions. This is the volume fraction of the original Stienen model. Here we consider some variants of these models: the dead leaves model with grains of a fixed convex shape and possibly random sizes and random orientations, and a generalisation of the Stienen model with convex grains growing at random speeds. The main result of this paper is that if the radius distribution in the dead leaves model equals the speed distribution in the Stienen model, then the volume fractions of the two models are the same in this case also. Furthermore, we show that for grains of a fixed shape and orientation, centrally symmetric sets give the highest volume fraction, while simplices give the lowest. If the grains are randomly rotated, then the volume fraction achieves its highest value only for spheres.

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.

Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread out’ in the sense of dilation than another one, X, if E(c(Y))≥E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Yʹ | Xʹ)=Xʹ, where (Xʹ,Yʹ) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts.