bluuebunny/crossref_metadata_2025 huggingface dataset:
A martingale characterization of Pólya-Lundberg processes
Authors: Niese Birgit
Published: July 2016

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.
Nonexponential asymptotics for the solutions of renewal equations, with applications
Authors: Yin Chuancun , Zhao Junsheng
Published: July 2016

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.
The correlation of the maxima of correlated Brownian motions
Authors: Rogers L. C. G. , Shepp Larry
Published: July 2016

We obtain an expression for the correlation of the maxima of two correlated Brownian motions.
Limiting crossing probabilities of random fields
Authors: Pereira L. , Ferreira H.
Published: July 2016

Random fields on , with long-range weak dependence for each coordinate individually, usually present clustering of high values. For each one of the eight directions in , we formulate restriction conditions on local occurrence of two or more crossings of high levels. These smooth oscillation conditions enable computation of the extremal index as a clustering measure from the limiting mean number of crossings. In fact, only four directions must be inspected since for opposite directions we find the same local path crossing behaviour and the same limiting mean number of crossings. The general theory is illustrated with several 1-dependent nonstationary random fields.
Conditioned stable Lévy processes and the Lamperti representation
Authors: Caballero M. E. , Chaumont L.
Published: July 2016

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.
A four-dimensional random motion at finite speed
Authors: Kolesnik Alexander D.
Published: July 2016

We consider the random motion of a particle that moves with constant finite speed in the space ℝ
4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position,
X
(t) = (X
1(t), X
2(t), X
3(t), X
4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(
x
, t), x ∈ ℝ
4, t ≥ 0, of
X
(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(
x
,t), of the absolutely continuous component of f(
x
,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.
Thou Shalt not Diversify: Why ‘Two Of Every Sort’?
Authors: Ibragimov Rustam
Published: July 2016

This paper presents a study of the intertemporal propagation of distributional properties of phenotypes in general polygenic multisex inheritance models with sex- and time-dependent heritabilities. It further analyzes the implications of these models under heavy-tailedness of traits' initial distributions. Our results suggest the optimality of a flexible asexual/binary mating system. Switching between asexual and binary inheritance mechanisms allows the population effectively to achieve a fast suppression of negative traits and a fast dispersion of positive traits, regardless of the distributional properties of the phenotypes in the initial period.
A Note on Insensitivity in Stochastic Networks
Authors: Zachary Stan
Published: July 2016

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).
Tail Asymptotics of the Supremum of a Regenerative Process
Authors: Palmowski Zbigniew , Zwart Bert
Published: July 2016

We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes involving both light tails and heavy tails, and are illustrated with examples arising in queueing theory and insurance risk.
On Long-Range Dependence in Regenerative Processes Based on a General ON/OFF Scheme
Authors: Leipus Remigijus , Surgailis Donatas
Published: July 2016

In this paper, we obtain a closed form for the covariance function of a general stationary regenerative process. It is used to derive exact asymptotics of the covariance function of stationary ON/OFF and workload processes, when ON and OFF periods are heavy-tailed and mutually dependent. The case of a G/G/1/0 queueing system with heavy-tailed arrival and/or service times is studied in detail.
On the Pricing of American Options in Exponential Lévy Markets
Authors: Ivanov Roman V.
Published: July 2016

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.
A Note on Planar Random Motion at Finite Speed
Authors: Kolesnik Alexander D.
Published: July 2016

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R
2, is given.
Robustness of Delta Hedging for Path-Dependent Options in Local Volatility Models
Authors: Schied Alexander , Stadje Mitja
Published: July 2016

We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.
Moments of Random Allocation Processes Reaching a Boundary
Authors: Tsitsiashvili G. S.
Published: July 2016

In this paper we develop some results presented by Gani (2004), deriving moments for random allocation processes. These moments correspond to the allocation processes reaching some domain boundary. Exact formulae for means, variances, and probability generating functions as well as some asymptotic formulae for moments of random allocation processes are obtained. A special choice of the asymptotics and of the domain allows us to reduce a complicated numerical procedure to a simple asymptotic one.
Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum
Authors: Zhou Xiaowen
Published: July 2016

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches S
τ(a) and I
κ(a), respectively. The main results of this paper concern the distributions of (τ(a), S
τ(a), τ−(a), Ŷ
τ(a)) and of (κ(a), I
κ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.
Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge
Authors: Lindell Andreas , Holst Lars
Published: July 2016

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.
Law of Large Numbers for Dynamic Bargaining Markets
Authors: Ferland René , Giroux Gaston
Published: July 2016

We describe the random meeting motion of a finite number of investors in markets with friction as a Markov pure-jump process with interactions. Using a sequence of these, we prove a functional law of large numbers relating the large motions with the finite market of the so-called continuum of agents.
Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution
Authors: Carlen Eric , Gabetta Ester , Regazzini Eugenio
Published: July 2016

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure onR. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets ofR). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variablesx̃1,x̃2, …, where these random variables have an infinite second moment and zero mean. Then, withTn:= ∑j=1ηnλj,nx̃j, with max1 ≤j≤ ηnλj,n→ 0 (asn→ +∞), and ∑j=1ηnλj,n2= 1,n= 1, 2, …, the classical central limit theorem suggests thatTshould in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.
The Large Deviation Principle for the On-Off Weibull Sojourn Process
Authors: Duffy Ken R. , Sapozhnikov Artem
Published: July 2016

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.
The Limit Behavior of Dual Markov Branching Processes
Authors: Li Yangrong , Pakes Anthony G. , Li Jia , Gu Anhui
Published: July 2016

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix
Q
is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.
Dynamic Distributed Scheduling in Random Access Networks
Authors: Stolyar Alexander L.
Published: July 2016

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.
A Lévy Insurance Risk Process with Tax
Authors: Albrecher Hansjörg , Renaud Jean-François , Zhou Xiaowen
Published: July 2016

Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy 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 Cramér-Lundberg risk model with tax.
A Compact Framework for Hidden Markov Chains with Applications to Fractal Geometry
Authors: Ruiz Víctor
Published: July 2016

We introduce a class of stochastic processes in discrete time with finite state space by means of a simple matrix product. We show that this class coincides with that of the hidden Markov chains and provides a compact framework for it. We study a measure obtained by a projection on the real line of the uniform measure on the Sierpinski gasket, finding that the dimension of this measure fits with the Shannon entropy of an associated hidden Markov chain.
On the Relationships Between Lumpability and Filtering of Finite Stochastic Systems
Authors: Ledoux James , White Langford B. , Brushe Gary D.
Published: July 2016

The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the approximation of a general FSS by a lumpable FSS.
Impact of Routeing on Correlation Strength in Stationary Queueing Network Processes
Authors: Daduna Hans , Szekli Ryszard
Published: July 2016

For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.
The Action Gambler and Equal-Sized Wagering
Authors: Hartvigsen David
Published: July 2016

A gambler with an initial bankroll is faced with a finite sequence of identical and independent bets. For each bet, he may wager up to his current bankroll, and will win this amount with probability p or lose it with probability 1-p. His problem is to devise a wagering strategy that will maximize his final expected utility with the side condition that the total amount wagered (i.e. the total ‘action’) be at least his initial bankroll. Our main result is an expression that characterizes when the strategy of placing equal-sized wagers on all bets is optimal. In particular, for a given bankroll B, utility function f (concave, increasing, differentiable), and n bets, we show that it is optimal to wager b/n on each bet if and only if the probability of winning each bet is less than or equal to some value p
⋆∈[½,1] (where p
⋆ is an explicit function of B, f, and n). We prove the result by using a basic nonlinear programming technique.
An Integral-Equation Approach for Defaultable Bond Prices with Application to Credit Spreads
Authors: Chen Yu-Ting , Lee Cheng-Few , Sheu Yuan-Chung
Published: July 2016

We study defaultable bond prices in the Black–Cox model with jumps in the asset value. The jump-size distribution is arbitrary, and following Longstaff and Schwartz (1995) and Zhou (2001) we assume that, if default occurs, the recovery at maturity depends on the ‘severity of default’. Under this general setting, the vehicle for our analysis is an integral equation. With the aid of this, we prove some properties of the bond price which are consistent numerically and empirically with earlier works. In particular, the limiting credit spread as time to maturity tends to 0 is nonzero. As a byproduct, we show that the integral equation implies an infinite-series expansion for the bond price.
Bounds for the Ruin Probability of a Discrete-Time Risk Process
Authors: Diasparra Maikol A. , Romera Rosario
Published: July 2016

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.
On the First Passage time for Brownian Motion Subordinated by a Lévy Process
Authors: Hurd T. R. , Kuznetsov A.
Published: July 2016

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.
Fixed Precision MCMC Estimation by Median of Products of Averages
Authors: Niemiro Wojciech , Pokarowski Piotr
Published: July 2016

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interest θ is represented as a product of expected values, θ = µ
1 ⋯ µ

k
, and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator , i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given for to have fixed relative precision at a given level of confidence, that is, to satisfy . Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.
On a Terminating Shock Process with Independent Wear Increments
Authors: Cha Ji Hwan , Finkelstein Maxim
Published: July 2016

In extreme shock models, only the impact of the current, possibly fatal shock is usually taken into account, whereas in cumulative shock models, the impact of the preceding shocks is accumulated as well. In this paper we combine an extreme shock model with a specific cumulative shock model. It is shown that the proposed setting can also be interpreted as a generalization of the well-known Brown–Proschan model that describes repair actions for repairable systems. For a system subject to a specific process of shocks, we derive the survival probability and the corresponding failure rate function. Some meaningful interpretations and examples are discussed.
On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts
Authors: Kella Offer
Published: July 2016

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.
Integrating Volatility Clustering Into Exponential Lévy Models
Authors: Bender Christian , Marquardt Tina
Published: July 2016

We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.
On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit
Authors: Covo Shai
Published: July 2016

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let X
ε be the sum of jumps not exceeding ε, and let µ(ε)=E[X
ε(1)]. We study the question of weak convergence of X
ε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of X
ε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).
Convolutions of Long-Tailed and Subexponential Distributions
Authors: Foss Sergey , Korshunov Dmitry , Zachary Stan
Published: July 2016

Convolutions of long-tailed and subexponential distributions play a major role in the analysis of many stochastic systems. We study these convolutions, proving some important new results through a simple and coherent approach, and also showing that the standard properties of such convolutions follow as easy consequences.
Bounds on Variances of Lifetimes of Coherent and Mixed Systems
Authors: Jasiński Krzysztof , Navarro Jorge , Rychlik Tomasz
Published: July 2016

We consider coherent and mixed reliability systems composed of elements with independent and identically distributed lifetimes. We present upper bounds on variances of system lifetimes, expressed in terms of variances of single components. We also discuss attainability conditions and some special cases and examples.
The Logarithmic Skew-Normal Distributions are Moment-Indeterminate
Authors: Lin Gwo Dong , Stoyanov Jordan
Published: July 2016

We study the class of logarithmic skew-normal (LSN) distributions. They have heavy tails; however, all their moments of positive integer orders are finite. We are interested in the problem of moments for such distributions. We show that the LSN distributions are all nonunique (moment-indeterminate). Moreover, we explicitly describe Stieltjes classes for some LSN distributions; they are families of infinitely many distributions, which are different but have the same moment sequence as a fixed LSN distribution.
Isomorphism and Symmetries in Random Phylogenetic Trees
Authors: Bóna Miklós , Flajolet Philippe
Published: July 2016

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.
A Three-Parameter Binomial Approximation
Authors: Peköz Erol A. , Röllin Adrian , Čekanavičius Vydas , Shwartz Michael
Published: July 2016

We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution, where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations are typically more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.
Optimal Stopping for Processes with Independent Increments, and Applications
Authors: Deligiannidis G. , Le H. , Utev S.
Published: July 2016

In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.
Stochastic Monotonicity and Continuity Properties of the Extinction Time of Bellman-Harris Branching Processes: An Application to Epidemic Modelling
Authors: González M. , Martínez R. , Slavtchova-Bojkova M.
Published: July 2016

The aim of this paper is to study the stochastic monotonicity and continuity properties of the extinction time of Bellman-Harris branching processes depending on their reproduction laws. Moreover, we show their applications in an epidemiological context, obtaining an optimal criterion to establish the proportion of susceptible individuals in a given population that must be vaccinated in order to eliminate an infectious disease. First the spread of infection is modelled by a Bellman-Harris branching process. Finally, we provide a simulation-based method to determine the optimal vaccination policies.
On the number of runs for Bernoulli arrays
Authors: Ait Aoudia Djilali , Marchand Éric
Published: July 2016

We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries X

k,j
and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of S

n
for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.
A class of location-independent variability orders, with applications
Authors: Shaked Moshe , Sordo Miguel A. , Suárez-Llorens Alfonso
Published: July 2016

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In this paper we study these orders when h is nondecreasing. We note that all these orders are location independent, and we characterize the dispersive order, and the location-independent riskier order, by means of the generalized TTT orders with nondecreasing h. Further properties, which strengthen known properties of the dispersive order, are given. A useful nontrivial closure property of the generalized TTT orders with nondecreasing h is obtained. Applications in poverty comparisons, risk management, and reliability theory are described.
Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model
Authors: Möhle M.
Published: July 2016

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chainXis studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix ofXand an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.
Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models
Authors: Lessard Sabin
Published: July 2016

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.
Sum the Multiplicative Odds to One and Stop
Authors: Tamaki Mitsushi
Published: July 2016

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.
Multiobjective Stopping Problem for Discrete-Time Markov Processes: Convex Analytic Approach
Authors: Dufour F. , Piunovskiy A. B.
Published: July 2016

The purpose of this paper is to study an optimal stopping problem with constraints for a Markov chain with general state space by using the convex analytic approach. The costs are assumed to be nonnegative. Our model is not assumed to be transient or absorbing and the stopping time does not necessarily have a finite expectation. As a consequence, the occupation measure is not necessarily finite, which poses some difficulties in the analysis of the associated linear program. Under a very weak hypothesis, it is shown that the linear problem admits an optimal solution, guaranteeing the existence of an optimal stopping strategy for the optimal stopping problem with constraints.
A Simple Model for Random Oscillations
Authors: Papangelou F.
Published: July 2016

A simple model for a randomly oscillating variable is suggested, which is a variant of the two-state random velocity model. As in the latter model, the variable keeps rising or falling with constant velocity for some time before randomly reversing its direction. In contrast however, its propensity to reverse depends on its current value and it is for this desirable feature that the model is proposed here. This feature has two implications: (a) neither the changing variable nor its velocity is Markovian, although the joint process is, and (b) the linear differential equations arising in the case of our model do not have constant coefficients. The results given in this paper are meant to illustrate the straightforward nature of some of the calculations involved and to highlight the relationship with one-dimensional diffusions.
Utility Optimization in Congested Queueing Networks
Authors: Walton N. S.
Published: July 2016

We consider a multiclass single-server queueing network as a model of a packet switching network. The rates packets are sent into this network are controlled by queues which act as congestion windows. By considering a sequence of congestion controls, we analyse a sequence of stationary queueing networks. In this asymptotic regime, the service capacity of the network remains constant and the sequence of congestion controllers act to exploit the network's capacity by increasing the number of packets within the network. We show that the stationary throughput of routes on this sequence of networks converges to an allocation that maximises aggregate utility subject to the network's capacity constraints. To perform this analysis, we require that our utility functions satisfy an exponential concavity condition. This family of utilities includes weighted α-fair utilities for α > 1.
Generalized Coupon Collection: The Superlinear Case
Authors: Smythe R. T.
Published: July 2016

We consider a generalized form of the coupon collection problem in which a random number,S, of balls is drawn at each stage from an urn initially containingnwhite balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after thekndraws? Our analysis is asymptotic asn→ ∞. We concentrate on the case whenkndraws are made, wherekn/n→ ∞ (the superlinear case), although we sketch known results for other ranges ofkn. A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.
Exact Lower Bounds on the Exponential Moments of Truncated Random Variables
Authors: Pinelis Iosif
Published: July 2016

Exact lower bounds on the exponential moments of min(y,X) andX1{X<y} are provided given the first two moments of a random variableX. 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 truncationX1{X<y} are demonstrated. An application to option pricing is given.
Conditional Full Support of Gaussian Processes with Stationary Increments
Authors: Gasbarra Dario , Sottinen Tommi , van Zanten Harry
Published: July 2016

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) ∼ c
1λ
p
e−c
2λ
q

for λ → ∞ (with necessarily p < 1 if q = 0), the CFS property holds if and only if q < 1.
Signatures of Coherent Systems Built with Separate Modules
Authors: Gertsbakh Ilya , Shpungin Yoseph , Spizzichino Fabio
Published: July 2016

The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.
Percolation of Words on Z d with Long-Range Connections
Authors: de Lima B. N. B. , Sanchis R. , Silva R. W. C.
Published: July 2016

Consider an independent site percolation model onZd, with parameterp∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameterp, there exists an integerK(p) such that all binary sequences (words) ξ ∈ {0, 1}Ncan be seen simultaneously, almost surely, even if all connections with length larger thanK(p) are suppressed. We also show some results concerning howK(p) should scale withpaspgoes to 0. Related results are also obtained for the question of whether or not almost all words are seen.
A revenue-equivalence theorem for electricity auctions
Authors: Hinz Juri
Published: July 2016

The purpose of this paper is to analyse the real-time trading of electricity. We address a model for an auction-like trading which captures key features of real-world electricity markets. Our main result establishes that, under certain conditions, the expected total payment for electricity is independent of the particular auction type. This result is analogous to the revenue-equivalence theorem known for classical auctions and could contribute to an improved understanding of different electricity market designs and their comparison.
Maximum dynamic entropy models
Authors: Asadi Majid , Ebrahimi Nader , Hamedani G. G. , Soofi Ehsan S.
Published: July 2016

A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.
Modelling long-range-dependent Gaussian processes with application in continuous-time financial models
Authors: Gao Jiti
Published: July 2016

This paper considers a class of continuous-time long-range-dependent Gaussian processes. The corresponding spectral density is assumed to have a general and flexible form, which covers some important and special cases. For example, the spectral density of a continuous-time fractional stochastic differential equation is included. A modelling procedure is then established through estimating the parameters involved in the spectral density by using an extended continuous-time version of the Gauss–Whittle objective function. The resulting estimates are shown to be strongly consistent and asymptotically normal. An application of the modelling procedure to the identification and modelling of a fractional stochastic volatility is discussed in some detail.
Optimal admission control for a single-server loss queue
Authors: Lin Kyle Y. , Ross Sheldon M.
Published: July 2016

This paper presents a single-server loss queueing system where customers arrive according to a Poisson process. Upon arrival, the customer presents itself to a gatekeeper who has to decide whether to admit the customer into the system without knowing the busy–idle status of the server. There is a cost if the gatekeeper blocks a customer, and a larger cost if an admitted customer finds the server busy and therefore has to leave the system. The goal of the gatekeeper is to minimize the total expected discounted cost on an infinite time horizon. In the case of an exponential service distribution, we show that a threshold-type policy—block for a time period following each admission and then admit the next customer—is optimal. For general service distributions, we show that a threshold-type policy need not be optimal; we then present a sufficient condition for the existence of an optimal threshold-type policy.
On Reliability Bounds via Conditional Inequalities
Authors: Xie M. , Lai C. D.
Published: July 2016

In this paper we study an approximation of system reliability using one-step conditioning. It is shown that, without greatly increasing the computational complexity, the conditional method may be used instead of the usual minimal cut and minimal path bounds to obtain more accurate approximations and bounds. We also study the conditions under which the approximations are bounds on the reliability. Some further extensions are also presented.
Bias Optimality in Controlled Queueing Systems
Authors: Haviv Moshe , Puterman Martin L.
Published: July 2016

This paper studies an admission control M/M/1 queueing system. It shows that the only gain (average) optimal stationary policies with gain and bias which satisfy the optimality equation are of control limit type, that there are at most two and, if there are two, they occur consecutively. Conditions are provided which ensure the existence of two gain optimal control limit policies and are illustrated with an example. The main result is that bias optimality distinguishes these two gain optimal policies and that the larger of the two control limits is the unique bias optimal stationary policy. Consequently it is also Blackwell optimal. This result is established by appealing to the third optimality equation of the Markov decision process and some observations concerning the structure of solutions of the second optimality equation.