Showing papers in "Journal of Applied Probability in 2008"
TL;DR: In this article, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived, which gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic.
Abstract: In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.
120 citations
TL;DR: In this paper, it was shown that the positive Wiener-Hopf factor of a Levy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation.
Abstract: We show that the positive Wiener-Hopf factor of a Levy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Levy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.
108 citations
TL;DR: In this paper, the reliability functions of residual lifetimes of used coherent systems under two particular conditions on the status of the components or the system in terms of the residual lifetime of order statistics are obtained.
Abstract: The representation of the reliability function of the lifetime of a coherent system as a mixture of the reliability function of order statistics associated with the lifetimes of its components is a very useful tool to study the ordering and the limiting behaviour of coherent systems. In this paper, we obtain several representations of the reliability functions of residual lifetimes of used coherent systems under two particular conditions on the status of the components or the system in terms of the reliability functions of residual lifetimes of order statistics.
106 citations
TL;DR: In this paper, the authors solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Levy risk process with tax payments of a loss-carry-forward type.
Abstract: Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Levy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected discounted aggregate income for the tax authority in this model. The results considerably generalise those for the Cramer-Lundberg risk model with tax.
95 citations
TL;DR: In this article, a robust numerical method to compute the scale function of a general spectrally negative Levy process (X, P) is presented, which is based on the Esscher transform of measure Pv. This change of measure makes it possible for the scale functions to be bounded and makes numerical computation easy, fast, and stable.
Abstract: In this paper we present a robust numerical method to compute the scale function W^ ix) of a general spectrally negative Levy process (X, P). The method is based on the Esscher transform of measure Pv under which X is taken and the scale function is determined. This change of measure makes it possible for the scale function to be bounded and, hence, makes numerical computation easy, fast, and stable. Working under the new measure Pv and using the method of Abate and Whitt (1992) and Choudhury, Lucantoni, and Whitt (1994), we give a fast stable numerical algorithm for the computation of W^q\x).
93 citations
TL;DR: Benaim and Friz as discussed by the authors showed that the tail of risk neutral returns can be related explicitly with the wing behavior of the Black-Scholes implied volatility smile, and obtained, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike.
Abstract: The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Levy models and examine several popular models in more detail, both analytically and numerically.
90 citations
TL;DR: In this article, the authors generalise some results for spectrally negative Levy processes to the setting of Markov additive processes (MAPs), where a prominent role is assumed by the first passage times, which are determined in terms of their Laplace transforms.
Abstract: The present paper generalises some results for spectrally negative Levy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.
78 citations
TL;DR: In this paper, the authors studied the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence.
Abstract: In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.
71 citations
TL;DR: The theory of language and automata is used to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and explicit constructive algorithms are given to build the PMC associated to any given pattern problem.
Abstract: In the framework of patterns in random texts, the Markov chain embedding techniques consist of turning the occurrences of a pattern over an order-m Markov sequence into those of a subset of states into an order-1 Markov chain. In this paper we use the theory of language and automata to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and we give explicit constructive algorithms to build the PMC associated to any given pattern problem. The interest of PMCs is then illustrated through the exact computation of P-values whose complexity is discussed and compared to other classical asymptotic approximations. Finally, we consider two illustrative examples of highly degenerated pattern problems (structured motifs and PROSITE signatures), which further illustrate the usefulness of our approach.
62 citations
TL;DR: In this article, the authors considered the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the distribution of X, and proved that P(Y ≤ t) = P(W(t) = 0).
Abstract: Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.
59 citations
TL;DR: In this paper, the authors consider an infectious disease spreading along the edges of a network which may have significant clustering and define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility.
Abstract: We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.
TL;DR: It is proved that two dynamic strategies, which are collectively call queue length based random access (QRA), ensure stability as long as the rates of exogenous arrival flows are within the network saturation rate region.
Abstract: We consider a model of random access (slotted-aloha-type) communication networks of general topology. Assuming that network links receive exogenous arrivals of packets for transmission, we seek dynamic distributed random access strategies whose goal is to keep all network queues stable. We prove that two dynamic strategies, which we collectively call queue length based random access (QRA), ensure stability as long as the rates of exogenous arrival flows are within the network saturation rate region. The first strategy, QRA-I, can be viewed as a random-access-model counterpart of the max-weight scheduling rule, while the second strategy, QRA-II, is a counterpart of the exponential (EXP) rule. The two strategies induce different dynamics of the queues in the fluid scaling limit, which can be exploited for the quality-of-service control in applications.
TL;DR: In this paper, the authors studied the -stable continuous state branching processes and the latter process conditioned never to become extinct in the light of positive self-similarity and provided explicit results concerning the paths of the two processes.
Abstract: In this paper we study the -stable continuous-state branching processes (for 2 (1,2]) and the latter process conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous state branching processes and the Lamperti transformation for positive self-similar Markov processes gives access to a number of explicit results concerning the paths of -stable continuousstate branching processes and -stable continuous-state branching processes conditioned never to become extinct.
TL;DR: In this paper, weak and strong survival for branching random walks on multigraphs with bounded degree was studied and it was shown that, at the strong critical value, the process dies out locally almost surely.
Abstract: We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.
TL;DR: In this article, a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions is presented, and the resulting distribution is called the PAREto copula.
Abstract: The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.
TL;DR: In this paper, the properties of total time on test transforms of order n and examine their applications in reliability analysis are studied. And the ageing properties of the baseline distribution is compared with those of transformed distributions, and a partial order based on ftth-order transforms and their implications are discussed.
Abstract: In this paper we study the properties of total time on test transforms of order n and examine their applications in reliability analysis. It is shown that the successive transforms produce either distributions with increasing or bathtub-shaped failure rates or distributions with decreasing or upside bathtub-shaped failure rates. The ageing properties of the baseline distribution is compared with those of transformed distributions, and a partial order based on ftth-order transforms and their implications are discussed.
TL;DR: In this paper, the genealogy of individuals with infinite descent in a supercritical continuous-state branching process is described, and the authors describe the family tree of individuals having infinite descent.
Abstract: We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.
TL;DR: In this article, stochastic monotone properties of the residual life and the inactivity time of k-out-of-n systems with independent and non-identically distributed components are discussed.
Abstract: By considering k-out-of-n systems with independent and nonidentically distributed components, we discuss stochastic monotone properties of the residual life and the inactivity time. We then present some stochastic comparisons of two systems based on the residual life and inactivity time.
TL;DR: In this paper, the number of collisions of an exchangeable coalescent with multiple collisions was studied and a coupling technique was used to derive limiting laws of X n, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval.
Abstract: We study the number of collisions, X n , of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite characteristic measure η(d x ) = x −2 Λ(d x ). Via a coupling technique, we derive limiting laws of X n , using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of X n include normal, stable with index 1 ≤ α a − 2, b ) distribution with parameters a > 2 and b > 0. The approach taken allows us to derive asymptotics of three other functionals of the coalescent: the absorption time, the length of an external branch chosen at random from the n external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.
TL;DR: In this paper, a transient analysis of the TCP window size process is presented, and the Laplace transform of the transient moments is derived for integer and fractional moments, as well as an explicit characterization of the speed of convergence to steady state.
Abstract: The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.
TL;DR: The negative binomial distribution is the most suitable approximate model for the number of k-runs as mentioned in this paper, and it outperforms the Poisson approximation, the general compound poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson-approximation in Rollin (2005).
Abstract: The distributions of the run occurrences for a sequence of independent and identically distributed (iid) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non iid experiments are generally intractable It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy In this paper we demonstrate that the negative binomial distribution is the most suitable approximate model for the number of k-runs: it outperforms the Poisson approximation, the general compound Poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson approximation in Rollin (2005) In particular, its accuracy of approximation in terms of the total variation distance improves when the number of experiments increases, in the same way as the normal approximation improves in the Berry-Esseen theorem
TL;DR: In this paper, the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and a family of such systems converges in law if and only if the corresponding empirical measure vectors converge in law.
Abstract: Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.
TL;DR: In this paper, the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation was studied.
Abstract: We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.
TL;DR: In this paper, a single-server queue with Levy input and a workload process (Q(t))(t >= 0), with a focus on the correlation structure, is considered, and it is shown that r(t) is positive, decreasing, and convex.
Abstract: In this paper we consider a single-server queue with Levy input and, in particular, its workload process (Q(t))(t >= 0), with a focus on the correlation structure. With the correlation function defined as r(t) := cov(Q(0), Q(1))/var(Q(0)) (assuming that the workload process is in stationarity at time 0), we first determine its transform integral(infinity)(0) r(t)e(-theta t) dt. This expression allows us to prove that r(.) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that r(.) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of r(t), for large t, for the cases of light-tailed and heavy-tailed Levy inputs.
TL;DR: In this article, the authors derived the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extended some of the results of Makri, Philippou and Psillakis (2007b).
Abstract: Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the P?lya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.
TL;DR: In this paper, the authors considered a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate when the surplus are negative.
Abstract: In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.
TL;DR: This work considers a reflected Lévy process without negative jumps, starting at the origin, and analyzes the steady-state distribution of the resulting process, reflected at theorigin.
Abstract: We consider a reflected Levy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Levy exponent of the Levy process is changed. As soon as the process hits zero again, the Levy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Levy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Levy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Levy process that is a subordinator until the timer expires.
TL;DR: In this paper, the authors apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay, and prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time.
Abstract: We study investment and disinvestment decisions in situations where there is a time lag d > 0 from the time t when the decision is taken to the time t + d when the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).
TL;DR: In this paper, the authors consider a swing contract between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values.
Abstract: Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.
TL;DR: In this paper, it was shown that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale and that its rate function is not convex.
Abstract: This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.