We consider a two-color Pólya urn in the case when a fixed number S of balls is added at each step. Assume it is a large urn that is, the second eigenvalue m of the replacement matrix satisfies 1∕2<m∕S≤1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order n^m∕S. The object of interest is the limit distribution of this random term.

The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree m. The limit laws appear to constitute a new family of probability densities supported by the whole real line.

This work considers a many-server queueing system in which customers with independent and identically distributed service times, chosen from a general distribution, enter service in the order of arrival. The dynamics of the system are represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterized as the unique solution to a coupled pair of integral equations which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, when the arrival process is time-homogeneous, the measure-valued component of the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.

We consider waves propagating in a randomly layered medium with long-range correlations. An example of such a medium is studied in [19] and leads, in particular, to an asymptotic travel time described in terms of a fractional Brownian motion. Here we study the asymptotic transmitted pulse under very general assumptions on the long-range correlations. In the framework that we introduce in this paper, we prove in particular that the asymptotic time-shift can be described in terms of non-Gaussian and/or multifractal processes.

Univariate superpositions of Ornstein–Uhlenbeck-type processes (OU), called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behavior. This paper introduces multivariate supOU processes and gives conditions for their existence and finiteness of moments. Moreover, the second-order moment structure is explicitly calculated, and examples exhibit the possibility of long-range dependence.

Our supOU processes are defined via homogeneous and factorizable Lévy bases. We show that the behavior of supOU processes is particularly nice when the mean reversion parameter is restricted to normal matrices and especially to strictly negative definite ones.

For finite variation Lévy bases we are able to give conditions for supOU processes to have locally bounded càdlàg paths of finite variation and to show an analogue of the stochastic differential equation of OU-type processes, which has been suggested in [2] in the univariate case. Finally, as an important special case, we introduce positive semi-definite supOU processes, and we discuss the relevance of multivariate supOU processes in applications.

Advective skew dispersion is a natural Markov process defined by a diffusion with drift across an interface of jump discontinuity in a piecewise constant diffusion coefficient. In the absence of drift, this process may be represented as a function of α-skew Brownian motion for a uniquely determined value of α=α^∗; see Ramirez et al. [Multiscale Model. Simul. 5 (2006) 786–801]. In the present paper, the analysis is extended to the case of nonzero drift. A determination of the (joint) distributions of key functionals of standard skew Brownian motion together with some associated probabilistic semigroup and local time theory is given for these purposes. An application to the dispersion of a solute concentration across an interface is provided that explains certain symmetries and asymmetries in recently reported laboratory experiments conducted at Lawrence–Livermore Berkeley Labs by Berkowitz et al. [Water Resour. Res. 45 (2009) W02201].

In this paper, we study simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they cannot be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings.

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point p_sd(q)=√q∕(1+√q), the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

We construct Zero-Coupon Bond markets driven by a cylindrical Brownian motion in which the notion of generalized portfolio has important flaws: There exist bounded smooth random variables with generalized hedging portfolios for which the price of their risky part is +∞ at each time. For these generalized portfolios, sequences of the prices of the risky part of approximating portfolios can be made to converges to any given extended real number in [−∞, ∞].

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where Y=(Y_t)_{t∈[0, 1]} is the solution of a multidimensional Lévy-driven stochastic differential equation and f is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions f that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the Lévy measure around zero.

In particular, one can derive upper bounds from the Blumenthal–Getoor index of the Lévy process. In the case where the Blumenthal–Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the Lévy process does not incorporate a Wiener process or if the Blumenthal–Getoor index β is larger than 4∕3, then the upper bound is of order τ^{−(4−β)∕(6β)} when the runtime τ tends to infinity. Whereas in the case, where β is in [1, 4∕3] and the Lévy process has a Gaussian component, we obtain bounds of order τ^{−β∕(6β−4)}. In particular, the error is at most of order τ^−1∕6.

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number n of elements, or a fraction of n or a logarithmic power of n.

We study the term structure equation for single-factor models that predict nonnegative short rates. In particular, we show that the price of a bond or a bond option is the unique classical solution to a parabolic differential equation with a certain boundary behavior for vanishing values of the short rate. If the boundary is attainable then this boundary behavior serves as a boundary condition and guarantees uniqueness of solutions. On the other hand, if the boundary is nonattainable then the boundary behavior is not needed to guarantee uniqueness but it is nevertheless very useful, for instance, from a numerical perspective.

We consider the branching random walks in d-dimensional integer lattice with time–space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a certain random variable. When d≥3 and the fluctuation of environment satisfies a certain uniform square integrability then it is nondegenerate and we prove a central limit theorem for the density of the population in terms of almost sure convergence.

For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter p_c the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at p_c. This behavior is often called “sharpness of the percolation transition.”

For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimensional contact-like processes to model vegetation patterns in an arid landscape (see [20]). In that context occupied clusters are interpreted as patches of vegetation. For some of these models it is reported in [20] that computer simulations indicate power-law behavior in some interval of positive length of a model parameter. This would mean that in these models the percolation transition is not sharp.

This motivated us to investigate similar questions for the ordinary (“basic”) 2D contact process with parameter λ. We show, using techniques from Bollobás and Riordan [8, 11], that for the upper invariant measure ν̄_λ of this process the percolation transition is sharp. If λ is such that (ν̄_λ-a.s.) there are no infinite clusters, then for all parameter values below λ the cluster-size distribution has exponential decay.