Sensitivity Analysis of Some Applied Probability Models

2000 Mathematics Subject Classi cation: Primary 90C31. Secondary 62C12, 62P05, 93C41.

Bounds for the probability generating functional of a Gibbs point process

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.

Context tree selection: A unifying view

Context tree models have been introduced by Rissanen in [25] as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics. The present paper investigates non-asymptotic properties of two popular procedures of context tree estimation: Rissanen's algorithm Context and penalized maximum likelihood. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning overestimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The under-estimation properties rely on classical hypotheses for processes of infinite memory. These results improve on and generalize the bounds obtained in Duarte et al. (2006) [12], Galves et al. (2008) [18], Galves and Leonardi (2008) [17], Leonardi (2010) [22], refining asymptotic results of Buhlmann and Wyner (1999) [4] and Csiszar and Talata (2006) [9]. (C) 2011 Elsevier B.V. All rights reserved.

A PHASE TRANSITION FOR COMPETITION INTERFACES

We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle theta of the cone: for theta >= 180 degrees, the direction is deterministic, while for theta < 180 degrees, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

POLLING SYSTEMS WITH PARAMETER REGENERATION, THE GENERAL CASE

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the sth moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447-1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space.

THE VANISHING DISCOUNT APPROACH FOR THE AVERAGE CONTINUOUS CONTROL OF PIECEWISE DETERMINISTIC MARKOV PROCESSES

This work is concerned with the existence of an optimal control strategy for the long-run average continuous control problem of piecewise-deterministic Markov processes (PDMPs). In Costa and Dufour (2008), sufficient conditions were derived to ensure the existence of an optimal control by using the vanishing discount approach. These conditions were mainly expressed in terms of the relative difference of the alpha-discount value functions. The main goal of this paper is to derive tractable conditions directly related to the primitive data of the PDMP to ensure the existence of an optimal control. The present work can be seen as a continuation of the results derived in Costa and Dufour (2008). Our main assumptions are written in terms of some integro-differential inequalities related to the so-called expected growth condition, and geometric convergence of the post-jump location kernel associated to the PDMP. An example based on the capacity expansion problem is presented, illustrating the possible applications of the results developed in the paper.

The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Similar Markov chains

enin et al. (2000) recently introduced the idea of similarity in the context of birth-death processes. This paper examines the extent to which their results can be extended to arbitrary Markov chains. It is proved that, under a variety of conditions, similar chains are strongly similar in a sense which is described, and it is shown that minimal chains are strongly similar if and only if the corresponding transition-rate matrices are strongly similar. A general framework is given for constructing families of strongly similar chains; it permits the construction of all such chains in the irreducible case.

Path integrals for continuous-time Markov chains

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martinez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Phi on the space of probability distributions on {1, 2,.. }. In the case of a birth-death process, the components of Phi(nu) can be written down explicitly for any given distribution nu. Using this explicit representation, we will show that Phi preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefevre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

Beyond Smith's rule: An optimal dynamic index, rule for single machine stochastic scheduling with convex holding costs

Most research on single machine scheduling has assumedthe linearity of job holding costs, which is arguablynot appropriate in some applications. This motivates ourstudy of a model for scheduling $n$ classes of stochasticjobs on a single machine, with the objective of minimizingthe total expected holding cost (discounted or undiscounted). We allow general holding cost rates that are separable,nondecreasing and convex on the number of jobs in eachclass. We formulate the problem as a linear program overa certain greedoid polytope, and establish that it issolved optimally by a dynamic (priority) index rule,whichextends the classical Smith's rule (1956) for the linearcase. Unlike Smith's indices, defined for each class, ournew indices are defined for each extended class, consistingof a class and a number of jobs in that class, and yieldan optimal dynamic index rule: work at each time on a jobwhose current extended class has larger index. We furthershow that the indices possess a decomposition property,as they are computed separately for each class, andinterpret them in economic terms as marginal expected cost rate reductions per unit of expected processing time.We establish the results by deploying a methodology recentlyintroduced by us [J. Niño-Mora (1999). "Restless bandits,partial conservation laws, and indexability. "Forthcomingin Advances in Applied Probability Vol. 33 No. 1, 2001],based on the satisfaction by performance measures of partialconservation laws (PCL) (which extend the generalizedconservation laws of Bertsimas and Niño-Mora (1996)):PCL provide a polyhedral framework for establishing theoptimality of index policies with special structure inscheduling problems under admissible objectives, which weapply to the model of concern.