Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.

We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted α-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556–567] [α∈(0, ∞)]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055–1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.

Under weighted proportional fair sharing of bandwidth (α=1) and a mild local traffic condition, we show how multiplicative state space collapse can be combined with uniqueness in law and an invariance principle for the diffusion [Theory Probab. Appl. 40 (1995) 1–40], [Ann. Appl. Probab. 17 (2007) 741–779] to establish a diffusion approximation for the workload process and hence to yield an approximation for the flow count process. In this case, the workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. When all of the weights are equal (proportional fair sharing), this diffusion has a product form invariant measure. If the latter is integrable, it yields the unique stationary distribution for the diffusion which has a strikingly simple interpretation in terms of independent dual random variables, one for each of the resources of the network. We are able to extend this product form result to the case where document sizes are distributed as finite mixtures of exponentials and to models that include multi-path routing. We indicate some difficulties related to extending the diffusion approximation result to values of α≠1.

We illustrate our approximation results for a few simple networks. In particular, for a two-resource linear network, the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity. For a four-resource network with multi-path routing, the product form result under proportional fair sharing is expressed in terms of independent dual random variables, one for each of a set of generalized cut constraints.

A zero-one sequence describes a path through a rooted directed binary tree T; it also encodes a real number in [0, 1]. We regard the level of the external node of T along the path as a function on the unit interval, the silhouette of T. We investigate the asymptotic behavior of the resulting stochastic processes for sequences of trees that are generated by the binary search tree algorithm.

We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity δ∈ℝ of the pinning interaction is constant, while the interface spacing T=T_N is allowed to vary with the size N of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large N limit, as a function of (T_N)_N and for fixed δ>0. In particular, we show that a transition occurs at T_N=O(log N). Our approach is based on renewal theory.

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.

In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice.

Here we provide a new point of view on this phenomenon. Using the gas construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we introduce a notion of convergence of graphs, such that if (G_n)→G then the gas process built on G_n converges in distribution to the gas process on G. That gives a general tool to show that gas processes related to animals enumeration are often Markovian on lines extracted from lattices.

We provide examples and computations of new generating functions for directed animals with various sources on the triangular lattice, on the $\mathcal{T}_{n}$ lattices introduced in [Ann. Comb. 4 (2000) 269–284] and on a generalization of the $\mathcal{L}_{n}$ lattices introduced in [J. Phys. A 29 (1996) 3357–3365].

We introduce a spatially explicit model for the competition between type a and type b alleles. Each vertex of the d-dimensional integer lattice is occupied by a diploid individual, which is in one of three possible states or genotypes: aa, ab or bb. We are interested in the long-term behavior of the gene frequencies when Mendel’s law of segregation does not hold. This results in a voter type model depending on four parameters; each of these parameters measures the strength of competition between genes during meiosis. We prove that with or without a spatial structure, type a and type b alleles coexist at equilibrium when homozygotes are poor competitors. The inclusion of a spatial structure, however, reduces the parameter region where coexistence occurs.

We introduce a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into seasons called alternately season 1 and season 2. We prove that there is an open set of the parameters for which both species can coexist when their dispersal range is large enough. Numerical simulations also suggest that three species can coexist in the presence of two seasons. This contrasts with the long-term behavior of the time-homogeneous multitype contact process for which the species with the higher birth rate outcompetes the other species when the death rates are equal.

The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here, we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. As an application, we can relate swap and quanto options to standard call and put options. Explicit calculations for jump models are also provided.

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarrès [Ann. Probab. (2007), to appear], that the attracting edge emerges with probability 1 whenever the underlying graph is locally bounded. We study the asymptotic behavior of the tail distribution of the (random) time of attraction. In particular, we obtain exact (up to a multiplicative constant) asymptotics if the underlying graph has two edges. Next, we show some extensions in the setting of finite graphs, and infinite graphs with bounded degree. As a corollary, we obtain the fact that if the reinforcement weight has the form w(k)=k^ρ, ρ>1, then (universally over finite graphs) the expected time to attraction is infinite if and only if $\rho\leq1+\frac{1+\sqrt{5}}{2}$.

One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking.

In this paper, we resolve this open problem, albeit in a somewhat unexpected way. We show that characterizing stable queueing networks is an algorithmically undecidable problem for the case of nonpreemptive static buffer priority scheduling policies and deterministic interarrival and service times. Thus, no constructive characterization of stable queueing networks operating under this class of policies is possible. The result is established for queueing networks with finite and infinite buffer sizes and possibly zero service times, although we conjecture that it also holds in the case of models with only infinite buffers and nonzero service times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002) 272–293] and uses the so-called counter machine device as a reduction tool.