This paper is concerned with various aspects of the simulation of one-dimensional reflected (or regulated) Brownian motion. The main result shows that the discretization error associated with the Euler scheme for simulation of such a process has both a strong and weak order of convergence of precisely 1/2. This contrasts with the faster order 1 achievable for simulations of SDE's without reflecting boundaries. The asymptotic distribution of the discretization error is described using Williams' decomposition of a Brownian path at the time of a minimum. Improved methods for simulation of reflected Brownian motion are discussed.

This paper provides an asymptotically efficient algorithm for the allocation of computing resources to the problem of Monte Carlo integration of continuous-time security prices. The tradeoff between increasing the number of time intervals per unit of time and increasing the number of simulations, given a limited budget of computer time, is resolved for first-order discretization schemes (such as Euler) as well as second- and higher-order schemes (such as those of Milshtein or Talay).

We characterize absence of arbitrage with tame portfolios in the case of invertible volatility matrix. As a corollary we get that, under a certain condition, absence of arbitrage with tame portfolios is characterized by the existence of the so-called equivalent martingale measure. Without that condition, the existence of equivalent martingale measure is equivalent to absence of approximate arbitrage. The proofs are probabilistic and are based on a construction of two specific arbitrages. Some examples are provided.

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes.

This paper treats transience for queueing network models by considering an associated fluid limit model. If starting from any initial condition the fluid limit model explodes at a linear rate, then the associated queueing network with i.i.d. service times and a renewal arrival process explodes faster than any fractional power.

We consider open queueing networks in which arrival and service rates are dependent on the state (i.e., queue length) of the network. They are modeled as multidimensional birth and death processes. If a heavy traffic condition is sastisfied on the behavior of arrival and service rates when the queue length becomes very large, it is shown that a properly normalized sequence of queue length converges in law to a reflecting diffusion process.

Choudhury and Lucantoni recently developed an algorithm for calculating moments of a probability distribution by numerically inverting its moment generating function. They also showed that high-order moments can be used to calculate asymptotic parameters of the complementary cumulative distribution function when an asymptotic form is assumed, such as $F^c(x) \sim \alpha x^\beta e^{-\eta x}$ as $x \rightarrow \infty$. Moment-based algorithms for computing asymptotic parameters are especially useful when the transforms are not available explicitly as in models of busy periods or polling systems. Here we provide additional theoretical support for this moment-based algorithm for computing asymptotic parameters and new refined estimators for the case $\beta \neq 0$. The new refined estimators converge much faster (as a function of moment order) than the previous estimators, which means that fewer moments are needed, thereby speeding up the algorithm. We also show how to compute all the parameters in a multiterm asymptote of the form $F^c(x) \sim \sum^m_{k = 1} \alpha_k x^{\beta - k + 1} e^{-\eta x}$. We identify conditions under which the estimators converge to the asymptotic parameters and we determine rates of convergence, focusing especially on the case $\beta \neq 0$. Even when $\beta = 0$, we show that it is necessary to assume the asymptotic form for the complementary distribution function; the asymptotic form is not implied by convergence of the moment-based estimators alone. In order to get good estimators of the asymptotic decay rate $\eta$ and the asymptotic power $\beta$ when $\beta \neq 0$, a multiple-term asymptotic expansion is required. Such asymptotic expansions typically hold when $\beta \neq 0$, corresponding to the dominant singularity of the transform being a multiple pole ($\beta$ a positive integer) or an algebraic singularity (branch point, $\beta$ noninteger). We also show how to modify the moment generating function in order to calculate asymptotic parameters when all moments do not exist (the case $\eta = 0)$.

A general (Crump-Mode-Jagers) spatial branching process is considered. The asymptotic behavior of the numbers present at time $t$ in sets of the form $\lbrack ta, \infty)$ is obtained. As a consequence it is shown that if $B_t$ is the position of the rightmost person at time $t, B_t/t$ converges to a constant, which can be obtained from the individual reproduction law, almost surely on the survival set of the process. This generalizes the known discrete-time results.

The stepping-stone model is widely used in population genetics to describe the evolution of a population with mating and geographical structure. It is typically formulated on a countable set, where each element of the set corresponds to a colony. Each colony consists of a population of a fixed number of haploid individuals. Individuals undergo random mating within each colony and migrate to neighboring colonies. In the model considered here, we are interested in the changes that may occur at a single locus on a single chromosome at which two alleles are possible. Changes in the gene frequencies are caused by migration between colonies, in addition to mutation, selection and random mating within each colony. It is frequently assumed that local populations persist indefinitely. This, however, is a reasonable assumption only if the average time until extinction of a local population is much larger than the time interval over which the whole population is studied. In this paper, using the framework of interacting particle systems, we address the question of how extinction and recolonization affect the spatial distribution of gene frequencies in both homogeneous and inhomogeneous environments. It turns out that extinction does not change the qualitative behavior of the model in a homogeneous environment; that is, in one and two dimensions, the population clusters in the absence of mutation (i.e., it consolidates into larger and larger blocks), whereas in higher dimensions, coexistence is possible. However, there are quantitative differences. For example, we find that extinction speeds up clustering in one dimension. In inhomogeneous environments, extinction can drastically change the behavior and new phenomena, such as coexistence in $d = 2$, appear.

Generalizing Sellke's construction, a general stochastic epidemic with non-Markovian transition behavior is considered. At time $t = 0$, the population of total size $K$ consists of $aK$ individuals that are infected by a certain disease (and infectious); the remaining $bK$ individuals are susceptible with respect to that disease. An initially susceptible individual $i$, when infected (call $A^K_i$ its time of infection), stays infectious for a period of length $r_i$, until it is removed. An initially infected individual $i$ stays infected for a period of length $\hat{r}_i$ until it is removed. Removed individuals can no longer be affected by the disease. A deterministic approximation as (as $K \rightarrow \infty$) to the empirical measure $\xi_K = \frac{1}{K} \sum^{aK}_{i=1} \delta_{(0,\hat{r}_i)} + \frac{1}{K} \sum^{bK}_{i=1} \delta_{(A^K_i, A^K_i + r_i)}$, describing the average path behavior, is established using Stein's method.

Consider a time-inhomogeneous Markov chain which converges in probability to a subset $S_0$ of its state space. Override the standard move mechanism up to a random transition time, almost surely finite, but not necessarily a stopping time. Under weak conditions, the modified process converges in probability to the same set $S_0$. Two examples of independent interest illustrate this result.

In this paper we present an asymptotic estimator, obtained by observing a noisy image, for the parameters of both a stationary Markov random field and an independent Bernoulli noise. We first estimate the parameter of the noise by solving a polynomial equation of moderate degree (about 6-7 in the one-dimensional Ising model and about 10-15 in the two-dimensional Ising model, for instance) and then apply the maximum pseudo-likelihood method after removing the noise. Our method requires no extra simulation and is likely to be applicable to any Markov random field, in any dimension. Here we present the general theory and some examples in one dimension; more interesting examples in two dimensions will be discussed at length in a companion paper.

The Navier-Stokes equation for the vorticity of a viscous and incompressible fluid in $\mathbf{R}^2$ is analyzed as a macroscopic equation for an underlying microscopic model of randomly moving vortices. We consider $N$ point vortices whose positions satisfy a stochastic ordinary differential equation on $\mathbf{R}^{2N}$, where the fluctuation forces are state dependent and driven by Brownian sheets. The state dependence is modeled to yield a short correlation length $\varepsilon$ between the fluctuation forces of different vortices. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) whose stochastic term is state dependent and small if $\varepsilon$ is small. Thereby we generalize the well known approach to the Euler equation to the viscous case. The solution is extended to a large class of signed measures conserving the total positive and negative vorticities, and it is shown to be a weak solution of the SNSE. For initial conditions in $L_2(\mathbf{R}^2, dr)$ the solutions are shown to live on the same space with continuous sample paths and an equation for the square of the $L_2$-norm is derived. Finally we obtain the macroscopic NSE as the correlation length $\varepsilon \rightarrow 0$ and $N \rightarrow \infty$ (macroscopic limit), where we assume that the initial conditions are sums of $N$ point measures. As a corollary to the above results we obtain the solution to a bilinear stochastic partial differential equation which can be interpreted as the temperature field in a stochastic flow.

We consider empirical spectral processes indexed by classes of functions for the case of stationary point processes. Conditions for the measurability and equicontinuity of these processes and a weak convergence result are established. The results can be applied to the spectral analysis of point processes. In particular, we discuss the application to parametric and nonparametric spectral density estimation.

Some existence and stability results for the equilibrium points of the one-dimensional Kohonen self-organizing neural network with two neighbors are extended to most nonincreasing neighborhood functions. All the functions mentioned in the neural literature are included. The assumption on the stimuli distribution is weakened, too. In the multidimensional setting, we derive from a general formula various stability and instability results.

General criteria are given to ensure that in a family of discrete random processes, given parameters exhibit convergence to the solution of a system of differential equations. As one application we consider random graph processes in which the maximum degree is bounded and show that the numbers of vertices of given degree exhibit this convergence as the total number of vertices tends to infinity. Two other applications are to random processes which generate independent sets of vertices in random $r$-regular graphs. In these cases, we deduce almost sure lower bounds on the size of independent sets of vertices in random $r$-regular graphs.

We consider a sequence matching problem involving the optimal alignment score for contiguous sequences, rewarding matches by one unit and penalizing for deletions and mismatches by parameters $\delta$ and $\mu$, respectively. Let $M_n$ be the optimal score over all possible choices of two contiguous regions. Arratia and Waterman conjectured that, when the score constant $a(\mu, \delta) < 0$, $P\big(\frac{M_n}{\log n} \rightarrow 2b\big) = 1$ for some constant $b$. Here we prove the conjecture affirmatively.