We study how the spread of computer viruses, worms and other self-replicating malware is affected by the logical topology of the network over which they propagate. We consider a model in which each host can be in one of 3 possible states—susceptible, infected or removed (cured and no longer susceptible to infection). We characterize how the size of the population that eventually becomes infected depends on the network topology. Specifically, we show that if the ratio of cure to infection rates is larger than the spectral radius of the graph, and the initial infected population is small, then the final infected population is also small in a sense that can be made precise. Conversely, if this ratio is smaller than the spectral radius, then we show in some graph models of practical interest (including power law random graphs) that the average size of the final infected population is large. These results yield insights into what the critical parameters are in determining virus spread in networks.

We study some recursive procedures based on exact or approximate Euler schemes with decreasing step to compute the invariant measure of Lévy driven SDEs. We prove the convergence of these procedures toward the invariant measure under weak conditions on the moment of the Lévy process and on the mean-reverting of the dynamical system. We also show that an a.s. CLT for stable processes can be derived from our main results. Finally, we illustrate our results by several simulations.

We investigate the connections between the mean pathwise regularity of stochastic processes and their L^r(ℙ)-functional quantization rates as random variables taking values in some L^p([0, T], dt)-spaces (0<p≤r). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (e.g., for Brownian motion or symmetric stable processes) so that the rate is optimal as a universal upper bound. As a first application, we establish the O((log N)^−1/2) upper bound for general Itô processes which include multidimensional diffusions. Then, we focus on the specific family of Lévy processes for which we derive a general quantization rate based on the regular variation properties of its Lévy measure at 0. The case of compound Poisson processes, which appear as degenerate in the former approach, is studied specifically: we observe some rates which are between the finite-dimensional and infinite-dimensional “usual” rates.

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663–714] which finds the Tracy–Widom limit for the largest eigenvalue of a nonsingular p-dimensional complex Wishart matrix W_ℂ(Ω_p, n) to the case of several of the largest eigenvalues of the possibly singular (n<p) matrix W_ℂ(Ω_p, n). As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643–1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion.

Using the constructed price systems, we show, under very general assumptions, the following “face-lifting” result: the asymptotic superreplication price of a European contingent claim g(S_T) equals ĝ(S₀), where ĝ is the concave envelope of g and S_t is the price of the asset at time t. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.

The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson–Dirichlet process PD(α, θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.

Suppose that local characteristics of several independent compound Poisson and Wiener processes change suddenly and simultaneously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential change detection in continuous time does not provide optimal rules for situations in which several marked count data and continuously changing signals are simultaneously observable. In this paper, optimal Bayesian sequential detection rules are developed for such problems when the marked count data is in the form of independent compound Poisson processes, and the continuously changing signals form a multi-dimensional Wiener process. An auxiliary optimal stopping problem for a jump-diffusion process is solved by transforming it first into a sequence of optimal stopping problems for a pure diffusion by means of a jump operator. This method is new and can be very useful in other applications as well, because it allows the use of the powerful optimal stopping theory for diffusions.

The paper is devoted to the study of nonlinear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born–Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.

In an incomplete Brownian-motion market setting, we propose a convex monotonic pricing functional for nonattainable bounded contingent claims which is compatible with prices for attainable claims. The pricing functional is defined as the convex conjugate of a generalized entropy penalty functional and an interpretation in terms of tracking with instantaneously vanishing risk can be given.

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener–Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation.

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization.

In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in “most” of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.

On a locally finite point set, a navigation defines a path through the point set from one point to another. The set of paths leading to a given point defines a tree known as the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on ℝ^d. We examine the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small-world graphs where new results are established.

We show that renormalized two-dimensional Lotka–Volterra models near criticality converge to a super-Brownian motion. This is used to establish long-term survival of a rare type for a range of parameter values near the voter model.

In earlier studies, the estimation of the volatility of a stock using information on the daily opening, closing, high and low prices has been developed; the additional information in the high and low prices can be incorporated to produce unbiased (or near-unbiased) estimators with substantially lower variance than the simple open–close estimator. This paper tackles the more difficult task of estimating the correlation of two stocks based on the daily opening, closing, high and low prices of each. If we had access to the high and low values of some linear combination of the two log prices, then we could use the univariate results via polarization, but this is not data that is available. The actual problem is more challenging; we present an unbiased estimator which halves the variance.