bluuebunny/crossref_metadata_2025 huggingface dataset:
Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes
Authors: Baurdoux E. J.
Published: July 2016

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).
Two Rationales Behind the ‘Buy-And-Hold or Sell-At-Once’ Strategy
Authors: Yam S. C. P. , Yung S. P. , Zhou W.
Published: July 2016

The trading strategy of ‘buy-and-hold for superior stock and sell-at-once for inferior stock’, as suggested by conventional wisdom, has long been prevalent in Wall Street. In this paper, two rationales are provided to support this trading strategy from a purely mathematical standpoint. Adopting the standard binomial tree model (or CRR model for short, as first introduced in Cox, Ross and Rubinstein (1979)) to model the stock price dynamics, we look for the optimal stock selling rule(s) so as to maximize (i) the chance that an investor can sell a stock precisely at its ultimate highest price over a fixed investment horizon [0,T]; and (ii) the expected ratio of the selling price of a stock to its ultimate highest price over [0,T]. We show that both problems have exactly the same optimal solution which can literally be interpreted as ‘buy-and-hold or sell-at-once’ depending on the value of p (the going-up probability of the stock price at each step): when p›½, selling the stock at the last time step N is the optimal selling strategy; when p=½, a selling time is optimal if the stock is sold either at the last time step or at the time step when the stock price reaches its running maximum price; and when p‹½, time 0, i.e. selling the stock at once, is the unique optimal selling time.
Double Kernel Estimation of Sensitivities
Authors: Elie Romuald
Published: July 2016

In this paper we address the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance, for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has recently been introduced in Elie, Fermanian and Touzi (2007) through a randomization of the parameter of interest combined with nonparametric estimation techniques. In this paper we study another type of estimator that turns out to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a slightly more stringent condition, its rate of convergence is the same as the one of the estimator introduced in Elie, Fermanian and Touzi (2007) and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new types of estimator for the sensitivities.
Markov Chains Conditioned Never to Wait Too Long at the Origin
Authors: Jacka Saul
Published: July 2016

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain,X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit asT→∞ in the cases where either the state space is finite orXis transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.
Asymptotics of Iterated Branching Processes
Authors: Piau Didier
Published: July 2016

Gaweł and Kimmel (1996) introduced and studied iterated Galton–Watson processes, (X

n
)
n≥0, possibly with thinning, as models of the number of repeats of DNA triplets during some genetic disorders. Our main results are the following. If the process indeed involves some thinning then extinction, {X

n
→0}, and explosion, {X

n
→∞}, can have positive probability simultaneously. If the underlying (simple) Galton–Watson process is nondecreasing with mean m then, conditionally on explosion, the ratios (log X

n+1
)/X

n
converge to logm almost surely. This simplifies the arguments of Gaweł and Kimmel, and confirms and extends a conjecture of Pakes (2003).
Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions
Authors: Li Haijun , Sun Yannan
Published: July 2016

The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.
Two-Sex Branching Processes with Offspring and Mating in a Random Environment
Authors: Ma S. , Molina M.
Published: July 2016

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.
The Vanishing Discount Approach for the Average Continuous Control of Piecewise Deterministic Markov Processes
Authors: Costa O. L. V. , Dufour F.
Published: July 2016

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 α-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.
On the Degree Sequence of an Evolving Random Graph Process and Its Critical Phenomenon
Authors: Wu Xian-Yuan , Dong Zhao , Liu Ke , Cai Kai-Yuan
Published: July 2016

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.
Coalescent theory for a Monoecious Random Mating Population with a Varying Size
Authors: Pollak Edward
Published: July 2016

Consider a monoecious diploid population with nonoverlapping generations, whose size varies with time according to an irreducible, aperiodic Markov chain with statesx1N,…,xKN, whereK≪N. It is assumed that all matings except for selfing are possible and equally probable. At time 0 a random sample ofn≪Ngenes is taken. Given two successive population sizesxjNandxiN, the numbers of gametes that individual parents contribute to offspring can be shown to be exchangeable random variables distributed asGij. Under minimal conditions on the first three moments ofGijfor alliandj, a suitable effective population sizeNeis derived. Then if time is recorded in a backward direction in units of 2Negenerations, it can be shown that coalescent theory holds.
A Damped Telegraph Random Process with Logistic Stationary Distribution
Authors: Di Crescenzo Antonio , Martinucci Barbara
Published: July 2016

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.
On Occurrences of F-S Strings in Linearly and Circularly Ordered Binary Sequences
Authors: Makri Frosso S.
Published: July 2016

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least)
k
1 + k
2, that is, (at least)
k
1 failures followed by (at least) k
2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k
1 + k
2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.
Optimal Multicommodity Flow Through the Complete Graph with Random Edge Capacities
Authors: Khandwawala Mustafa , Sundaresan Rajesh
Published: July 2016

We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximum utility, given by the average of a concave function of each commodity flow, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.
Optimal Allocation of Active Redundancies to k-out-of-n Systems with Heterogeneous Components
Authors: Li Xiaohu , Ding Weiyong
Published: July 2016

In this note we deal with the allocation of independent and identical active redundancies to a k-out-of-n system with the usual stochastic order among its independent components. The optimal policy is proved both to assign more redundancies to the weaker component and to majorize all other policies. This improves the corresponding one in Hu and Wang (2009) and serves as a nice supplement to that in Misra, Dhariyal and Gupta (2009) as well.
A quintuple law for Markov additive processes with phase-type jumps
Authors: Breuer Lothar
Published: July 2016

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.
Preferential duplication graphs
Authors: Cohen Netta , Jordan Jonathan , Voliotis Margaritis
Published: July 2016

We consider a preferential duplication model for growing random graphs, extending previous models of duplication graphs by selecting the vertex to be duplicated with probability proportional to its degree. We show that a special case of this model can be analysed using the same stochastic approximation as for vertex-reinforced random walks, and show that ‘trapping’ behaviour can occur, such that the descendants of a particular group of initial vertices come to dominate the graph.
Approximation for expectations of unbounded functions of dependent integer-valued random variables
Authors: Ruzankin Pavel S.
Published: July 2016

Expectations of unbounded functions of dependent nonnegative integer-valued random variables are approximated by the expectations of the functions of independent copies of these random variables. The Lindeberg method is used.
Nonidentifiability of the Two-State Markovian Arrival Process
Authors: Ramírez-Cobo Pepa , Lillo Rosa E. , Wiper Michael P.
Published: July 2016

In this paper we consider the problem of identifiability for the two-state Markovian arrival process (MAP2). In particular, we show that the MAP2 is not identifiable, providing the conditions under which two different sets of parameters induce identical stationary laws for the observable process.
Asymptotic Normality of Discrete-Time Markov Control Processes
Authors: Mendoza-Pérez Armando F. , Hernández-Lerma Onésimo
Published: July 2016

In this paper we study the asymptotic normality of discrete-time Markov control processes in Borel spaces, with possibly unbounded cost. Under suitable hypotheses, we show that the cost sequence is asymptotically normal. As a special case, we obtain a central limit theorem for (noncontrolled) Markov chains.
Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates
Authors: Block Henry W. , Langberg Naftali A. , Savits Thomas H. , Wang Jie
Published: July 2016

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.
The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables
Authors: Chen Yiqing , Chen Anyue , Ng Kai W.
Published: July 2016

A sequence of random variables is said to be extended negatively dependent (END) if the tails of its finite-dimensional distributions in the lower-left and upper-right corners are dominated by a multiple of the tails of the corresponding finite-dimensional distributions of a sequence of independent random variables with the same marginal distributions. The goal of this paper is to establish the strong law of large numbers for a sequence of END and identically distributed random variables. In doing so we derive some new inequalities of large deviation type for the sums of END and identically distributed random variables being suitably truncated. We also show applications of our main result to risk theory and renewal theory.
A Markov Chain Analysis of Genetic Algorithms: Large Deviation Principle Approach
Authors: Suzuki Joe
Published: July 2016

In this paper we prove that the stationary distribution of populations in genetic algorithms focuses on the uniform population with the highest fitness value as the selective pressure goes to ∞ and the mutation probability goes to 0. The obtained sufficient condition is based on the work of Albuquerque and Mazza (2000), who, following Cerf (1998), applied the large deviation principle approach (Freidlin-Wentzell theory) to the Markov chain of genetic algorithms. The sufficient condition is more general than that of Albuquerque and Mazza, and covers a set of parameters which were not found by Cerf.
A Markov Model for the Spread of Viruses in an Open Population
Authors: Coutin L. , Decreusefond L. , Dhersin J. S.
Published: July 2016

Inspired by methods of queueing theory, we propose a Markov model for the spread of viruses in an open population with an exogenous flow of infectives. We apply it to the diffusion of AIDS and hepatitis C diseases among drug users. From a mathematical point of view, the difference between the two viruses is shown in two parameters: the probability of curing the disease (which is 0 for AIDS but positive for hepatitis C) and the infection probability, which seems to be much higher for hepatitis. This model bears some resemblance to the M/M/∞ queueing system and is thus rather different from the models based on branching processes commonly used in the epidemiological literature. We carry out an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a nonlinear autonomous differential system. We prove both a law of large numbers and a functional central limit theorem to determine the speed of convergence towards the limiting system. The deterministic system itself converges, as time tends to ∞, to an equilibrium point. We then show that the sequence of stationary probabilities of the stochastic models shrinks to a Dirac measure at this point. This means that in a large population and for long-term analysis, we may replace the individual-based microscopic stochastic model with the macroscopic deterministic system without loss of precision. Moreover, we show how to compute the sensitivity of any functional of the Markov process with respect to a slight variation of any parameter of the model. This approach is applied to the spread of diseases among drug users, but could be applied to many other case studies in epidemiology.
First Passage of a Markov Additive Process and Generalized Jordan Chains
Authors: D‘Auria Bernardo , Ivanovs Jevgenijs , Kella Offer , Mandjes Michel
Published: July 2016

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.
Random Networks with Preferential Growth and Vertex Death
Authors: Deijfen Maria
Published: July 2016

A dynamic model for a random network evolving in continuous time is defined, where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a functionbof the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a functiondof its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution {pk} is derived and analyzed for a number of specific choices ofbandd. Whenb(i) =i+ α andd(i) = β, that is, linear preferential attachment for the newborn and random deaths, thenpk∼k-(2+α). Whenb(i) =i+ 1 andd(i) = β(i+ 1), with β < 1, thenpk∼ (1 + β)−k, that is, if the death rate is also proportional to the fitness, then the power-law distribution is lost. Furthermore, whenb(i) =i+ 1 andd(i) = β(i+ 1)γ, with β, γ < 1, then logpk∼ -kγ, a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.
Matchmaking and Testing for Exponentiality in the M/G/∞ Queue
Authors: Grübel Rudolf , Wegener Hendrik
Published: July 2016

Customers arrive sequentially at times x
1 < x
2 < · · · < x

n
and stay for independent random times Z
1, …, Z

n
> 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times x

i
+ Z

i
are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.
On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant
Authors: Lee Chihoon
Published: July 2016

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R
+

d
, with drift r
0 ∈ R

d
and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r
0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, sup
x∈B
E
x
[τ
B
(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τ
B
(δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.
Managing Queues with Heterogeneous Servers
Authors: Kim Jung Hyun , Ahn Hyun-Soo , Righter Rhonda
Published: July 2016

We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.
Conditionally Independent Increment Point Processes
Authors: Vélez Ibarrola Ricardo , Prieto-Rumeau Tomás
Published: July 2016

In this paper we introduce conditionally independent increment point processes, that is, processes that are conditionally independent inside and outside a bounded set A given N(A), the number of points in A. We show that these point processes can be characterized by means of the avoidance function of a multinomial ‘support process’, the solution of a suitably defined linear system of equations, and, finally, the infinitesimal matrix of a continuous-time Markov chain.
On the Growth of the One-Dimensional Reverse Immunization Contact Processes
Authors: Tzioufas A.
Published: July 2016

We are concerned with the variation of the supercritical nearest-neighbours contact process such that first infection occurs at a lower rate; it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and, thus, obtain the corresponding strong law and central limit theorem. We also extend complete convergence in this case.
A Consistent Markov Partition Process Generated from the Paintbox Process
Authors: Crane Harry
Published: July 2016

We study a family of Markov processes onP(k), the space of partitions of the natural numbers with at mostkblocks. The process can be constructed from a Poisson point process onR+x ∏i=1kP(k)with intensity dt⊗ ϱν(k), where ϱνis the distribution of the paintbox based on the probability measure ν onPm, the set of ranked-mass partitions of 1, and ϱν(k)is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.
Micropulses and Different Types of Brownian Motion
Authors: Marouby Matthieu
Published: July 2016

In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.
A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant
Authors: Lee Chihoon
Published: July 2016

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+

d
, with drift r
0 ∈ ℝ
d
and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r
0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E
x
[V(Ž̆(1))] − V(x) ≤ −βV(x) + b
1

C
(x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.
Signature-Based Representations for the Reliability of Systems with Heterogeneous Components
Authors: Navarro Jorge , Samaniego Francisco J. , Balakrishnan N.
Published: July 2016

Signature-based representations of the reliability functions of coherent systems with independent and identically distributed component lifetimes have proven very useful in studying the ageing characteristics of such systems and in comparing the performance of different systems under varied criteria. In this paper we consider extensions of these results to systems with heterogeneous components. New representation theorems are established for both the case of components with independent lifetimes and the case of component lifetimes under specific forms of dependence. These representations may be used to compare the performance of systems with homogeneous and heterogeneous components.
Distribution of Clump Statistics for a Collection of Words
Authors: Martin Donald E. K. , Coleman Deidra A.
Published: July 2016

We give an efficient method based on minimal deterministic finite automata for computing the exact distribution of the number of occurrences and coverage of clumps (maximal sets of overlapping words) of a collection of words. In addition, we compute probabilities for the number ofh-clumps, word groupings where gaps of a maximal lengthhbetween occurrences of words are allowed. The method facilitates the computation ofp-values for testing procedures. A word is allowed to contain other words of the collection, making the computation more general, but also more difficult. The underlying sequence is assumed to be Markovian of an arbitrary order.
Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law
Authors: Khmaladze Estáte V.
Published: July 2016

Letxdenote a vector of lengthqconsisting of 0s and 1s. It can be interpreted as an ‘opinion’ comprised of a particular set of responses to a questionnaire consisting ofqquestions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered bynindividuals, thus providingn‘opinions’. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2qdifferent opinions, what number, μn, would one expect to see in the sample? How many of these opinions, μn(k), will occur exactlyktimes? In this paper we give an asymptotic expression for μn/ 2qand the limit for the ratios μn(k)/μn, when the number of questionsqincreases along with the sample sizenso thatn= λ2q, where λ is a constant. Letp(x) denote the probability of opinionx. The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensitiesnp(x). For example, one of our results states that, under certain natural conditions, for anyz> 0, ∑1{np(x) >z}=dnz−u,dn=o(2q).
The stochastic precedence ordering with applications in sampling and testing
Authors: Boland Philip J. , Singh Harshinder , Cukic Bojan
Published: July 2016

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.
On the maximum drawdown of a Brownian motion
Authors: Magdon-Ismail Malik , Atiya Amir F. , Pratap Amrit , Abu-Mostafa Yaser S.
Published: July 2016

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).
Backward SDEs with two barriers and continuous coefficient: an existence result
Authors: Lepeltier Jean-Pierre , San Martín Jaime
Published: July 2016

In this work we prove the existence of a solution for a doubly reflected backward SDE with a continuous linearly increasing coefficient in the case where the barriers L and U are such that L < U on [0,T) and there exists a continuous semimartingale between L and U.
Separable lower triangular bilinear model
Authors: Wang Hai-Bin , Wei Bo-Cheng
Published: July 2016

The aim of this paper is to analyze the probabilistic structure for a rather general class of bilinear models systematically. First, the sufficient and necessary conditions for stationarity are given with a concise expression. Then both the autocovariance function and the spectral density function are obtained. The Yule–Walker-type difference equations for autocovariances are derived by means of the spectral density function. Concerning the second-order probabilistic structure, the model is similar to an ARMA model. The third-order probabilistic structure for the model is discussed and a group of Yule–Walker-type difference equations for third-order cumulants are discovered.
Minimal martingale measures for jump diffusion processes
Authors: Arai Takuji
Published: July 2016

We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.
A note on the upper bound for the optimal work size
Authors: Cha Ji Hwan
Published: July 2016

Mi (2002) recently considered a two-dimensional optimization problem for the optimal age-replacement policy and the optimal work size. In order to find (y
∗,T
∗), Mi (2002) found the optimal age-replacement policy T
∗(y) for each fixed work size y, and then searched for the optimal work size y
∗. When applying this approach, for each fixed work size y, Mi (2002) obtained the bounds for T
∗(y). However, no bound for the optimal work size y
∗ was derived. In this note, the results on the upper bound for the optimal work size y
∗ are given.
On the role played by extreme summands when a sum of independent and identically distributed random vectors is asymptotically α-stable
Authors: Davydov Yu. , Nagaev A. V.
Published: July 2016

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.
Decomposition property in a discrete-time queue with multiple input streams and service interruptions
Authors: Ishizaki Fumio
Published: July 2016

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.
The Equations of Markovian Random Evolution on the Line
Authors: Kolesnik Alexander
Published: July 2016

We consider a general model of one-dimensional random evolution with n velocities and rates of a switching Poisson process (n ≥ 2). A governing nth-order hyperbolic equation in a determinant form is given. For two important particular cases it is written in an explicit form. Some known hyperbolic equations are obtained as particular cases of the general model
General Conditions for Comparing the Reliability Functions of Systems of Components Sharing a Common Environment
Authors: Garren Steven T. , Richards Donald St. P.
Published: July 2016

We present general criteria for analyzing the crossing characteristics of R
I, the reliability function of an m-of-n system of components operating within a laboratory (or test-bench) environment, and R
O, the reliability function of the same system now operating subject to an external environment. Inside the laboratory the components' lifetimes may be dependently distributed, and the external environment is modeled using the general approach of Lindley and Singpurwalla (1986). Our techniques, which utilize results basic to the theory of order statistics, apply to broad classes of external environment models.
Partial Orderings of Distributions Based on Right-Spread Functions
Authors: Fernandez-Ponce J. M. , Kochar S. C. , Muñoz-Perez J.
Published: July 2016

In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive ordering and it retains most of its interesting properties.
Characterizations of the Exponential Distribution via the Blocking Time in a Queueing System with an Unreliable Server
Authors: Xu Jian-Lun
Published: July 2016

The characterization of the exponential distribution via the coefficient of the variation of the blocking time in a queueing system with an unreliable server, as given by Lin (1993), is improved by substantially weakening the conditions. Based on the coefficient of variation of certain random variables, including the blocking time, the normal service time and the minimum of the normal service and the server failure times, two new characterizations of the exponential distribution are obtained.
First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths
Authors: Di Crescenzo Antonio
Published: July 2016

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.
Globally convergent stochastic optimization with optimal asymptotic distribution
Authors: Dippon Jürgen
Published: July 2016

A stochastic gradient descent method is combined with a consistent auxiliary estimate to achieve global convergence of the recursion. Using step lengths converging to zero slower than 1/n and averaging the trajectories, yields the optimal convergence rate of 1/√n and the optimal variance of the asymptotic distribution. Possible applications can be found in maximum likelihood estimation, regression analysis, training of artificial neural networks, and stochastic optimization.
Robustness results for the coalescent
Authors: Möhle M.
Published: July 2016

A variety of convergence results for genealogical and line-of-descendent processes are known for exchangeable neutral population genetics models. A general convergence-to-the-coalescent theorem is presented, which works not only for a larger class of exchangeable models but also for a large class of non-exchangeable population models. The coalescence probability, i.e. the probability that two genes, chosen randomly without replacement, have a common ancestor one generation backwards in time, is the central quantity to analyse the ancestral structure.
Recursive estimation of distributional fix-points
Authors: Embrechts Paul , Walk Harro
Published: July 2016

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L
1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).
Stochastic differential portfolio games
Authors: Browne Sid
Published: July 2016

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.
Poisson limits and nonparametric estimation for pairwise interaction point processes
Authors: Chang Wei-Bin , Gubner John A.
Published: July 2016

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.
Optimal control of a dam using P
λ,τ
M policies and penalty cost when the input process is a compound Poisson process with positive drift
Authors: Abdel-Hameed Mohamed
Published: July 2016

In this paper we consider the optimal control of an infinite dam using policies assuming that the input process is a compound Poisson process with a non-negative drift term, and using the total discounted cost and long-run average cost criteria. The results of Lee and Ahn (1998) as well as other well-known results are shown to follow from our results.
Asymptotic properties of integral functionals of geometric stochastic processes
Authors: Csáki Endre , Csörgő Miklós , Földes Antónia , Révész Pál
Published: July 2016

We study strong asymptotic properties of two types of integral functionals of geometric stochastic processes. These integral functionals are of interest in financial modelling, yielding various option pricings, annuities, etc., by appropriate selection of the processes in their respective integrands. We show that under fairly general conditions on the latter processes the logs of the integral functionals themselves asymptotically behave like appropriate sup functionals of the processes in the exponents of their respective integrands. We illustrate the possible use and applications of these strong invariance theorems by listing and elaborating on several examples.
An unexpected connection between branching processes and optimal stopping
Authors: Assaf David , Goldstein Larry , Samuel-Cahn Ester
Published: July 2016

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z

n
with offspring distribution Y, there exists a random variable X such that the probability P(Z

n
= 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X
1,…, X

n
, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.
Open-loop routeing to M parallel servers with no buffers
Authors: Altman Eitan , Bhulai Sandjai , Gaujal Bruno , Hordijk Arie
Published: July 2016

In this paper we study the assignment of packets to M parallel heterogeneous servers with no buffers. The controller does not have knowledge on the state of the servers and bases all decisions on the number of time slots ago that packets have been sent to the different servers. The objective of the controller is to minimize the expected average cost. We consider a general stationary arrival process, with no independence assumptions. We show that the problem can be transformed into a Markov Decision Process (MDP). We further show under quite general conditions on cost functions that only a finite number of states have to be considered in this MDP, which then implies the optimality of periodic policies. For the case of two servers we obtain a more detailed structure of the cost and optimal policies. In particular we show that the average cost function is multimodular, and we obtain expressions for the cost for MMPP and MAP processes. Finally we present an application to optimal robot scheduling for Web search engines.
Zero-temperature ising spin dynamics on the homogeneous tree of degree three
Authors: Howard C. Douglas
Published: July 2016

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Ising model on the homogeneous tree of degree three (𝕋) with random (i.i.d. Bernoulli) spin configuration at time 0. Letting θ denote the probability that any particular vertex has a +1 initial spin, for infinite spin clusters do not exist at time 0 but we show that infinite ‘spin chains’ (doubly infinite paths of vertices with a common spin) exist in abundance at any time ϵ > 0. We study the structure of the subgraph of 𝕋 generated by the vertices in time-ϵ spin chains. We show the existence of a phase transition in the sense that, for some critical θ
c
with spin chains almost surely never form for θ < θc
but almost surely do form in finite time for θ > θc
. We relate these results to certain quantities of physical interest, such as the t → ∞ asymptotics of the probability that any particular vertex changes spin after time t.