This paper gives a refinement of the results of Hunt and Kurtz on the dynamical behavior of large loss networks. We introduce a Liapounov function technique which, under the limiting regime of Kelly, enables the unique identification of limiting dynamics in many applications. This technique considerably simplifies much previous work in this area. We further apply it to the study of the dynamical behavior of large single-resource loss systems under virtual partitioning, or dynamic trunk reservation, controls. We identify limiting dynamics under the above regime, describing the behavior of the number of calls of each type in the system. We show that all trajectories of these dynamics converge to a single fixed point, which we identify. We also identify limiting stationary behavior, including call acceptance probabilities.

Cumulative broadband network traffic is often thought to be well modeled by fractional Brownian motion (FBM). However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable Lévy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence.

We study the tail behavior of the distribution of certain subadditive functionals acting on the sample paths of Lévy processes. The functionals we consider have, roughly speaking, the following property: only the points of the process that lie above a certain curve contribute to the value of the functional. Our assumptions will make sure that the process ends up eventually below the curve. Our results apply to ruin probabilities, distributions of sojourn times over curves, last hitting times and other functionals.

In this paper we consider large $\theta$ approximations for the stationary distribution of the neutral infinite alleles model as described by the the Poisson–Dirichlet distribution with parameter $\theta$. We prove a variety of Gaussian limit theorems for functions of the population frequencies as the mutation rate $\theta$ goes to infinity. In particular, we show that if a sample of size $n$ is drawn from a population described by the Poisson–Dirichlet distribution, then the conditional probability of a particular sample configuration is asymptotically normal with mean and variance determined by the Ewens sampling formula. The asymptotic normality of the conditional sampling distribution is somewhat surprising since it is a fairly complicated function of the population frequencies. Along the way, we also prove an invariance principle giving weak convergence at the process level for powers of the size-biased allele frequencies.

We consider the supercritical bisexual Galton–Watson process (BGWP) with promiscuous mating, that is, a branching process which behaves like an ordinary supercritical Galton–Watson process (GWP) as long as at least one male is born in each generation. For a certain example, it was pointed out by Daley, Hull and Taylor [J. Appl. Probab. 23 (1986) 585–600] that the extinction probability of such a BGWP apparently behaves like a constant times the respective probability of its asexual counterpart (where males do not matter) if the number of ancestors grows to $\infty$. In an earlier paper, we provided general upper and lower bounds for the ratio between both extinction probabilities and also numerical results that seemed to confirm the convergence of that ratio. However, theoretical considerations rather led us to the conjecture that this does not generally hold. The present article turns this conjecture into a rigorous result. The key step in our analysis is to identify the extinction probability ratio as a certain functional of a subcritical ordinary GWP and to prove its continuity as a function of the number of ancestors in a suitable topology associated with the entrance Martin boundary of that GWP.

We consider a controlled process governed by $X^{x, \theta} = x + \int \theta dS + H^{\theta}$, where $S$ is a semimartingale, $\Theta$ the set of control processes . is a convex subset of $L(S)$ and ${H^{\theta} :\theta \in \Theta}$ is a concave family of adapted processes with finite variation. We study the problem of minimizing the shortfall risk defined as the expectation of the shortfall $(B - X_T^{x, \theta})_+$ weighted by some loss function, where $B$ is a given nonnegative measurable random variable. Such a criterion has been introduced by Föllmer and Leukert [Finance Stoch. 4 (1999) 117–146] motivated by a hedging problem in an incomplete financial market context:$\Theta = L(S)$ and $H^{\theta} \equiv 0$. Using change of measures and optional decomposition under constraints, we state an existence result to this optimization problem and show some qualitative properties of the associated value function. A verification theorem in terms of a dual control problem is established which is used to obtain a quantitative description of the solution. Finally, we give some applications to hedging problems in constrained portfolios, large investor and reinsurance models.

We consider an optimal investment problem for a factor model treated by Bielecki and Pliska (Appl. Math. Optim. 39 337–360) as a risk-sensitive stochastic control problem, where the mean returns of individual securities are explicitly affected by economic factors defined as Gaussian processes. We relax the measurability condition assumed as Bielecki and Pliska for the investment strategies to select. Our investment strategies are supposed to be chosen without using information of factor processes but by using only past information of security prices. Then our problem is formulated as a kind of stochastic control problem with partial information. The case on a finite time horizon is discussed by Nagai (Stochastics in Finite and Infinite Dimension 321–340. Birkhäuser, Boston). Here we discuss the problem on infinite time horizon.

In mathematical finance, the price of the so-called “American Put option” is given by the value function of the optimal-stopping problem with the option payoff $\psi: x \to (K - x)^+$ as a reward function. Even in the Black–Scholes model, no closed-formula is known and numerous numerical approximation methods have been specifically designed for this problem.

In this paper, as an application of the theoretical result of B. Jourdain and C. Martini [Ann. Inst. Henri Poincaré Anal. Nonlinear 18 (2001) 1–17], we explore a new approximation scheme: we look for payoffs as close as possible to $\psi$, the American price of which is given by the European price of another claim. We exhibit a family of payoffs $\hat{\varphi}_h$ indexed by a measure $h$, which are continuous, match with $(K - x)^+$ outside of the range $]K_*, K[$ (where $K_*$ is the perpetual Put strike), are analytic inside with the right derivative ( -1) at both ends. Moreover a numerical procedure to select the best $h$ in some sense yields nice results.

In this paper we consider Foster–Liapounov-type drift conditions for Markov chains which imply polynomial rate convergence to stationarity in appropriate V-norms. We also show how these results can be used to prove central limit theorems for functions of the Markov chain. We consider two examples concerning random walks on the half line and the independence sampler.

This article studies a stochastic model of an evolutionary algorithm that evolves a “population” of potential solutions to a minimization problem. The minimization process is based on two operators. First, each solution is regarded as an individual that attempts a random search on a graph, involving a probabilistic operator called exploration. The second operator is called selection. This deterministic operator creates interaction between individuals. The convergence of the evolutionary process is described within the framework of simulated annealing. It can be quantified by means of two quantities called the critical height and the optimal convergence exponent, which both measure the difficulty of the algorithm to deal with the minimization problem. This work describes the critical height for large enough population sizes. Explicit bounds are given for the optimal convergence exponent, using a few geometric quantities. As an application, this work allows comparisons of the evolutionary strategy with independent parallel runs of the simulated annealing algorithm, and it helps deciding when one method should be preferred to the other.

Consider sequential packing of unit balls in a large cube, as in the Rényi car-parking model, but in any dimension and with finite input. We prove a law of large numbers and central limit theorem for the number of packed balls in the thermodynamic limit. We prove analogous results for numerous related applied models, including cooperative sequential adsorption, ballistic deposition, and spatial birth-growth models.

The proofs are based on a general law of large numbers and central limit theorem for “stabilizing” functionals of marked point processes of independent uniform points in a large cube, which are of independent interest. “Stabilization” means, loosely, that local modifications have only local effects.

We propose a method for numerical approximation of backward stochastic differential equations. Our method allows the final condition of the equation to be quite general and simple to implement. It relies on an approximation of Brownian motion by simple random walk.

A topical operator on $\mathbb{R}^d$ is one which is isotone and homogeneous. Let ${A(n) : n \geq 1}$ be a sequence of i.i.d. random topical operators such that the projective radius of $A(n) \dots A(1)$ is almost surely bounded for large $n$. If ${x(n) : n \geq 1}$, is a sequence of vectors given by $x(n) = A(n) \dots A(1)x_0$, for some fixed initial condition $x_0$, then the sequence ${x(n)/n : n \geq 1}$ satisfies a weak large deviation principle. As corollaries of this result we obtain large deviation principles for products of certain random aperiodic max-plus and min-plus matrix operators and for products of certain random aperiodic nonnegative matrix operators.

I this paper, we are interested in approxximation th entropy solution of a one-dimensional inviscid scalar conservation law starting from an initial condition with bounded variation owing to a system of interacting diffusions. We modify the system of signed particles associated with the parabolic equation obtained from the addition of a viscous term to this equation by killing couples of particles with opposite sign that merge. The sample paths of the corresponding reordered particles can be seen as probabilistic characteristic along which the approximate solution is constant. This enables us to prove that when the viscosity vanishes as the initial number of particles goes to $+\infty$, the approximate solution converges to the unique entropy solution of the inviscid conservation law. We illustrate this convergence by numerical results.

This paper is concerned with a class of nonlinear stochastic wave equations in $\mathbb{R}^d$ with $d \leq 3$, for which the nonlinear terms are polynomial of degree $m$. As an example of the nonexistence of a global solution in general, it is shown that there exists an explosive solution of some cubically nonlinear wave equation with a noise term. Then the existence and uniqueness theorems for local and global solutions in Sobolev space $H_1$ are proven with the degree of polynomial $m \leq 3$ for $d = 3$, and $m \geq 2$ for $d = 1$ or 2.

Let $\{(X_n, S_n): n = 0, 1, \dots\}$ be a Markov additive process, where $\{X_n\}$ is a Markov chain on a general state space and $S_n$ is an additive component on $\mathbb{R}^d$. We consider $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ as $\varepsilon \to 0$, where $A \subset \mathbb{R}^d$ is open and the mean drift of $\{S_n\}$ is away from $A$. Our main objective is to study the simulation of $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ using the Monte Carlo technique of importance sampling. If the set $A$ is convex, then we establish (i) the precise dependence (as $\varepsilon \to 0$) of the estimator variance on the choice of the simulation distribution and (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of D. Siegmund [Ann. Statist. 4 (1976) 673-684]. We then extend our techniques to the case where $A$ is not convex. Our results lead to positive conclusions which complement the multidimensional counterexamples of P. Glasserman and Y. Wang [Ann. Appl. Probab. 7 (1997) 731-746].