We consider the level hitting times τ_y = inf{t ≥ 0 | X_t = y} and the running maximum process M_t = sup{X_s | 0 ≤ s ≤ t} of a growth-collapse process (X_t)_t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random `collapse' times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τ_y can be determined in terms of the extended generator of X_t and give a power series expansion of the reciprocal of Ee^-sτ_y. We prove asymptotic results for τ_y and M_t: for example, if m(y) = Eτ_y is of rapid variation then M_t / m^-1(t) →^w 1 as t → ∞, where m^-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and X_t is ergodic, then M_t / m^-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

We present a number of new solutions to an integral equation arising in the limiting theory of Bellman-Harris processes. The argument proceeds via straightforward analysis of Mellin transforms. We also derive a criterion for the analyticity of the Laplace transform of the limiting distribution on Re(u) ≥ -c for some c > 0.

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In the associated Fleming-Viot process N particles evolve independently in Λ with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 / N.

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T^d. In the discrete torus N^-1T^d_N, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λ^c, the exchange rate is 1; if one site is in Λ and the other site is in Λ^c, and the direction of the bond connecting the sites is e_j, then the exchange rate is defined as N^-1 times the absolute value of the inner product between e_j and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.

Nonspatial stochastic models of populations subject to catastrophic events result in the common conclusion that the survival probability of the population is nondecreasing with respect to the random number of individuals removed at each catastrophe. The purpose of this paper is to prove that such a monotonic relationship is not true for simple spatial models based on Harris' contact processes, whose dynamics are described by hypergraph structures rather than traditional graph structures. More precisely, it is proved that, for a wide range of parameters, the destruction of (infinite) hyperplanes does not affect the existence of a nontrivial invariant measure, whereas the destruction of large (finite) cubes drives the population to extinction, a result that we depict by using the biological picture: forest fires are more devastating than tornadoes. This indicates that the geometry of the subsets struck by catastrophes is somewhat more important than their area, thus the need to consider spatial rather than nonspatial models in this context.

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l_n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l_n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

Improved bounds on the copula of a bivariate random vector are computed when partial information is available, such as the values of the copula on a given subset of [0, 1]², or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x₀ > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe^6βt] for positive constants k and p for times t exceeding some positive value u. We obtain explicit expressions for k, p, and u in terms of β, σ, x₀, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.

Web servers have to be protected against overload since overload can lead to a server breakdown, which in turn causes high response times and low throughput. In this paper, a stochastic model for breakdowns of server systems due to overload is proposed and an admission control policy which protects Web servers by controlling the amount and rate of work entering the system is studied. Requests from the clients arrive at the server following a nonhomogeneous Poisson process and each requested job takes a random time to be completed. It is assumed that the breakdown rate of the server depends on the number of jobs which are currently being performed by the server. Based on the proposed model, the reliability function and the breakdown rate function of the server system are derived. Furthermore, the long-run expected number of jobs completed per unit time is derived as the efficiency measure, and the optimal admission control policy which maximizes the efficiency will be discussed.

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, say b. In great generality such a probability converges to 0 exponentially fast in a power of b. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (in b) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hitting b in finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number, n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations as n ↗ ∞. A remarkable feature of TBS is that it is not based on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

In this paper we introduce conditionally independent increment point processes, that is, processes that are conditionally independent inside and outside a bounded set A given N(A), the number of points in A. We show that these point processes can be characterized by means of the avoidance function of a multinomial `support process', the solution of a suitably defined linear system of equations, and, finally, the infinitesimal matrix of a continuous-time Markov chain.

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

Let X_n be a sequence of integrable real random variables, adapted to a filtration (G_n). Define C_n = √{(1 / n)∑_k=1ⁿX_k - E(X_n+1 | G_n)} and D_n = √n{E(X_n+1 | G_n) - Z}, where Z is the almost-sure limit of E(X_n+1 | G_n) (assumed to exist). Conditions for (C_n, D_n) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑_k=1ⁿX_k - Z} = C_n + D_n → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.

Exact lower bounds on the exponential moments of min(y, X) and X1{X < y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y, X) over the truncation X1{X < y} are demonstrated. An application to option pricing is given.

We investigate the conditional full support (CFS) property, introduced in Guasoni et al. (2008a), for Gaussian processes with stationary increments. We give integrability conditions on the spectral measure of such a process which ensure that the process has CFS or not. In particular, the general results imply that, for a process with spectral density f such that f(λ) ∼ c₁λ^pe^-c₂λq for λ → ∞ (with necessarily p < 1 if q = 0), the CFS property holds if and only if q < 1.

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

We investigate the maximal number M_k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k^-1M_k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.

Let {M_n}_n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form Q_A(x) = lim_n→∞P(M_n ≤ x | M₀ ≤ A, M₁ ≤ A, ..., M_n ≤ A). Suppose that M₀ has distribution Q_A, and define T_A^Q_A = min{n | M_n > A, n ≥ 1}, the first time when M_n exceeds A. We provide sufficient conditions for Q_A(x) to be nonincreasing in A (for fixed x) and for T_A^Q_A to be stochastically nondecreasing in A.