bluuebunny/crossref_metadata_2025 huggingface dataset:
Dependent Risk Models with Bivariate Phase-Type Distributions
Authors: Badescu Andrei L. , Cheung Eric C. K. , Landriault David
Published: July 2016

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.
Bipower Variation for Gaussian Processes with Stationary Increments
Authors: Barndorff-Nielsen Ole E. , Corcuera José Manuel , Podolskij Mark , Woerner Jeannette H. C.
Published: July 2016

Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.
Optimal Stopping of a Brownian Bridge
Authors: Ekström Erik , Wanntorp Henrik
Published: July 2016

We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian bridge. Our examples constitute natural finite-horizon optimal stopping problems with explicit solutions.
Blackwell Optimality for Controlled Diffusion Processes
Authors: Jasso-Fuentes Héctor , Hernández-Lerma Onésimo
Published: July 2016

In this paper we study m-discount optimality (m≥ −1) and Blackwell optimality for a general class of controlled (Markov) diffusion processes. To this end, a key step is to express the expected discounted reward function as a Laurent series, and then search certain control policies that lexicographically maximize themth coefficient of this series form= −1,0,1,…. This approach naturally leads tom-discount optimality and it gives Blackwell optimality in the limit asm→ ∞.
A Two-Step Branching Splitting Model Under Cost Constraint for Rare Event Analysis
Authors: Lagnoux-Renaudie Agnès
Published: July 2016

In this paper we consider the splitting method first introduced in rare event analysis. In this technique, the sample paths are split into R multiple copies at various stages to speed up the simulation. Given the cost, the optimization of the algorithm suggests taking all the transition probabilities to be equal; nevertheless, in practice, these quantities are unknown. We address this problem by presenting an algorithm in two phases.
A Weak Limit Theorem for Generalized Jiřina Processes
Authors: Li Yuqiang
Published: July 2016

In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.
Similar States in Continuous-Time Markov Chains
Authors: Yap V. B.
Published: July 2016

In a homogeneous continuous-time Markov chain on a finite state space, two states that jump to every other state with the same rate are called similar. By partitioning states into similarity classes, the algebraic derivation of the transition matrix can be simplified, using hidden holding times and lumped Markov chains. When the rate matrix is reversible, the transition matrix is explicitly related in an intuitive way to that of the lumped chain. The theory provides a unified derivation for a whole range of useful DNA base substitution models, and a number of amino acid substitution models.
Strong Laws for Balanced Triangular Urns
Authors: Bose Arup , Dasgupta Amites , Maulik Krishanu
Published: July 2016

Consider an urn model whose replacement matrix is triangular, has all nonnegative entries, and the row sums are all equal to 1. We obtain strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use these strong laws to study further behavior of certain three-color urn models.
A Stochastic Model for Phylogenetic Trees
Authors: Liggett Thomas M. , Schinazi Rinaldo B.
Published: July 2016

We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical, we obtain a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical, we obtain a phylogenetic tree consistent with an HIV tree (many types at any given time, none lasting very long).
The Stationary Distributions of Two Classes of Reflected Ornstein–Uhlenbeck Processes
Authors: Xing Xiaoyu , Zhang Wei , Wang Yongjin
Published: July 2016

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.
Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models
Authors: Huillet Thierry , Möhle Martin
Published: July 2016

A Markov chainXwith finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions fromitoi-1 occur with probability (i/N)(1-p(i/N)) and transitions fromitoi+1 occur with probability (1-i/N)p(i/N). Herep:[0,1]→[0,1] is a given function. It is shown that ifpis continuous withp(x)≤p(1) for allx∈[0,1] then, for eachN, a dual processYtoX(with respect to a specific duality function) exists if and only if 1-pis completely monotone withp(0)=0. A probabilistic interpretation ofYin terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions onp, the processY, properly time and space scaled, converges to an Ornstein–Uhlenbeck process asNtends to ∞. The asymptotics of the stationary distribution ofYis studied asNtends to ∞. Examples are presented involving selection mechanisms. results.
Does Waste Recycling Really Improve the Multi-Proposal Metropolis–Hastings algorithm? an Analysis Based on Control Variates
Authors: Delmas Jean-Françcois , Jourdain Benjamin
Published: July 2016

The waste-recycling Monte Carlo (WRMC) algorithm introduced by physicists is a modification of the (multi-proposal) Metropolis–Hastings algorithm, which makes use of all the proposals in the empirical mean, whereas the standard (multi-proposal) Metropolis–Hastings algorithm uses only the accepted proposals. In this paper we extend the WRMC algorithm to a general control variate technique and exhibit the optimal choice of the control variate in terms of the asymptotic variance. We also give an example which shows that, in contradiction to the intuition of physicists, the WRMC algorithm can have an asymptotic variance larger than that of the Metropolis–Hastings algorithm. However, in the particular case of the Metropolis–Hastings algorithm called the Boltzmann algorithm, we prove that the WRMC algorithm is asymptotically better than the Metropolis–Hastings algorithm. This last property is also true for the multi-proposal Metropolis–Hastings algorithm. In this last framework we consider a linear parametric generalization of WRMC, and we propose an estimator of the explicit optimal parameter using the proposals.
Gaps in Discrete Random Samples
Authors: Grübel Rudolf , Hitczenko Paweł
Published: July 2016

Let (X

i
)
i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0
of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X
1,…,X

n
} of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q

k
)
k∈ℕ0
,
q

k
=P(X
1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑
k=0
∞
q

k+1/q

k
<∞ and limk→∞
q

k+1/q

k
=0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q

k+1/q

k
→ 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.
Optimal Choice of the Best Available Applicant in Full-Information Models
Authors: Tamaki Mitsushi
Published: July 2016

The problem we consider here is a full-information best-choice problem in which n applicants appear sequentially, but each applicant refuses an offer independently of other applicants with known fixed probability 0≤q<1. The objective is to maximize the probability of choosing the best available applicant. Two models are distinguished according to when the availability can be ascertained; the availability is ascertained just after the arrival of the applicant (Model 1), whereas the availability can be ascertained only when an offer is made (Model 2). For Model 1, we can obtain the explicit expressions for the optimal stopping rule and the optimal probability for a given n. A remarkable feature of this model is that, asymptotically (i.e. n→∞), the optimal probability becomes insensitive to q and approaches 0.580 164. The planar Poisson process (PPP) model provides more insight into this phenomenon. For Model 2, the optimal stopping rule depends on the past history in a complicated way and seems to be intractable. We have not solved this model for a finite n but derive, via the PPP approach, a lower bound on the asymptotically optimal probability.
General tax Structures and the Lévy Insurance Risk Model
Authors: Kyprianou Andreas E. , Zhou Xiaowen
Published: July 2016

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.
An Alternative Condition for Stochastic Domination
Authors: Hosaka Kenshi
Published: July 2016

We will propose an alternative condition for stochastic domination. This condition differs in an essential way from the strong likelihood ratio property. We also show an example which satisfies the new condition, but does not satisfy the strong likelihood ratio property.
Tail Asymptotics for a Random Sign Lindley Recursion
Authors: Vlasiou Maria , Palmowski Zbigniew
Published: July 2016

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
Two New Models for the Two-Person Red-And-Black Game
Authors: Chen May-Ru , Hsiau Shoou-Ren
Published: July 2016

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.
Stein's Method for Compound Geometric Approximation
Authors: Daly Fraser
Published: July 2016

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.
On the Variety of Shapes on the Fringe of a Random Recursive Tree
Authors: Feng Qunqiang , Mahmoud Hosam M.
Published: July 2016

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.
Generalized Measures of Divergence in Survival Analysis and Reliability
Authors: Vonta Filia , Karagrigoriou Alex
Published: July 2016

Measures of divergence or discrepancy are used either to measure mutual information concerning two variables or to construct model selection criteria. In this paper we focus on divergence measures that are based on a class of measures known as Csiszár's divergence measures. In particular, we propose a measure of divergence between residual lives of two items that have both survived up to some time t as well as a measure of divergence between past lives, both based on Csiszár's class of measures. Furthermore, we derive properties of these measures and provide examples based on the Cox model and frailty or transformation model.
Busy periods in fluid queues with multiple emptying input states
Authors: Field A. J. , Harrison P. G.
Published: July 2016

A semi-numerical method is derived to compute the Laplace transform of the equilibrium busy period probability density function in a fluid queue with constant output rate when the buffer is nonempty. The input process is controlled by a continuous-time semi-Markov chain (CTSMC) with n states such that in each state the input rate is constant. The holding time in states with net positive output rate - so-called emptying states - is assumed to be an exponentially distributed random variable, whereas in states with net positive input rate - so-called filling states - it may have an arbitrary probability distribution. The result is demonstrated by applying it to various systems, including fluid queues with two on-off input sources. The latter exercise in part shows consistency with prior results but also solves the problem in the case where there are two emptying states. Numerical results are presented for selected examples which expose discontinuities in the busy period distribution when the number of emptying states changes, e.g. as a result of increasing the fluid arrival rate in one or more states of the controlling CTSMC.
Energy efficiency of consecutive fragmentation processes
Authors: Fontbona Joaquín , Krell Nathalie , Martínez Servet
Published: July 2016

Motivated by a problem arising in the mining industry, we present a first study of the energy required to reduce a unit mass fragment by consecutively using several devices. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced in Bertoin and Martínez (2005), we compute the average energy required to attain a size η0 with this two-device procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that, for a certain range of parameters of the fragmentation processes and of their energy cost functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or vice versa.
Nonhomogeneous random walks systems on ℤ
Authors: Lebensztayn Elcio , Machado Fábio Prates , Zuluaga Martinez Mauricio
Published: July 2016

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.
Optimal liquidation of a call spread
Authors: Ekström Erik , Lindberg Carl , Tysk Johan , Wanntorp Henrik
Published: July 2016

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.
A note on converging geometric-type processes
Authors: Finkelstein Maxim
Published: July 2016

The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed.
Stochastic Integrals and Conditional Full Support
Authors: Pakkanen Mikko S.
Published: July 2016

We present conditions that imply the conditional full support (CFS) property, introduced in Guasoni, Rásonyi and Schachermayer (2008), for processes Z := H + ∫K dW, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case, under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.
Branching Process Approach for 2-Sat Thresholds
Authors: Mossel Elchanan , Sen Arnab
Published: July 2016

It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability α
i
/ 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.
Total Variation Approximation for Quasi-Stationary Distributions
Authors: Barbour A. D. , Pollett P. K.
Published: July 2016

Quasi-stationary distributions, as discussed in Darroch and Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.
Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes
Authors: Fu James C. , Wu Tung-Lung
Published: July 2016

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.
Standby Redundancy Allocations in Series and Parallel Systems
Authors: Misra Neeraj , Misra Amit Kumar , Dhariyal Ishwari Dutt
Published: July 2016

To enhance the performance of a system, a common practice employed by reliability engineers is to use redundant components in the system. In this paper we compare lifetimes of series (parallel) systems arising out of different allocations of one or two standby redundancies. These comparisons are made with respect to the increasing concave (convex) order, the hazard rate order, and the stochastic precedence order. The main results extend some related conclusions in the literature.
Hazard Rate Properties of a General Counting Process Stopped at an Independent Random Time
Authors: Badia F. G.
Published: July 2016

In this work we provide sufficient conditions under which a general counting process stopped at a random time independent from the process belongs to the reliability decreasing reversed hazard rate (DRHR) or increasing failure rate (IFR) class. We also give some applications of these results in generalized renewal and trend renewal processes stopped at a random time.
Asymptotic Fluid Optimality and Efficiency of the Tracking Policy for Bandwidth-Sharing Networks
Authors: Avrachenkov Konstantin , Piunovskiy Alexey , Zhang Yi
Published: July 2016

Optimal control of stochastic bandwidth-sharing networks is typically difficult. In order to facilitate the analysis, deterministic analogues of stochastic bandwidth-sharing networks, the so-called fluid models, are often taken for analysis, as their optimal control can be found more easily. The tracking policy translates the fluid optimal control policy back to a control policy for the stochastic model, so that the fluid optimality can be achieved asymptotically when the stochastic model is scaled properly. In this work we study the efficiency of the tracking policy, that is, how fast the fluid optimality can be achieved in the stochastic model with respect to the scaling parameter. In particular, our result shows that, under certain conditions, the tracking policy can be as efficient as feedback policies.
The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics
Authors: Spencer Simon E. F. , O‘Neill Philip D.
Published: July 2016

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.
Further Calculations for the McKean Stochastic Game for a Spectrally Negative Lévy Process: From a Point to an Interval
Authors: Baurdoux E. J. , Van Schaik K.
Published: July 2016

Following Baurdoux and Kyprianou (2008) we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Lévy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular, we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and ‘thickens’ to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump diffusion case are provided.
A Technique for Computing the PDFs and CDFs of Nonnegative Infinitely Divisible Random Variables
Authors: Veillette Mark S. , Taqqu Murad S.
Published: July 2016

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX
= e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.
On the Skewness of Order Statistics in Multiple-Outlier Models
Authors: Kochar Subhash , Xu Maochao
Published: July 2016

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the generalk-out-of-nsystems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.
Stochastic Comparison of Discounted Rewards
Authors: Righter Rhonda
Published: July 2016

It is well know that the expected exponentially discounted total reward for a stochastic process can also be defined as the expected total undiscounted reward earned before an independent exponential stopping time (let us call this the stopped reward). Feinberg and Fei (2009) recently showed that the variance of the discounted reward is smaller than the variance of the stopped reward. We strengthen this result to show that the discounted reward is smaller than the stopped reward in the convex ordering sense.
New Limiting Distributions for Bellman-Harris Processes
Authors: Angerer Wolfgang P.
Published: July 2016

We present a number of new solutions to an integral equation arising in the limiting theory of Bellman-Harris processes. The argument proceeds via straightforward analysis of Mellin transforms. We also derive a criterion for the analyticity of the Laplace transform of the limiting distribution on Re(u) ≥ -c for some c > 0.
On the Markov Transition Kernels for First Passage Percolation on the Ladder
Authors: Schlemm Eckhard
Published: July 2016

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l

n
between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l

n
/ n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.
Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options
Authors: Tankov Peter
Published: July 2016

Improved bounds on the copula of a bivariate random vector are computed when partial information is available, such as the values of the copula on a given subset of [0, 1]2, or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.
Efficient Simulation for the Maximum of Infinite Horizon Discrete-Time Gaussian Processes
Authors: Blanchet Jose , Li Chenxin
Published: July 2016

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, sayb. In great generality such a probability converges to 0 exponentially fast in a power ofb. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (inb) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hittingbin finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number,n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations asn↗ ∞. A remarkable feature of TBS is that it isnotbased on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.
Asymptotic Properties of a Leader Election Algorithm
Authors: Kalpathy Ravi , Mahmoud Hosam M. , Ward Mark Daniel
Published: July 2016

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.
Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model
Authors: Cai Ning
Published: July 2016

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.
On Durbin's Series for the Density of First Passage Times
Authors: Zipkin P.
Published: July 2016

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.
First Passage Times of (Reflected) Ornstein-Uhlenbeck Processes Over Random Jump Boundaries
Authors: Bo Lijun , Wang Yongjin , Yang Xuewei
Published: July 2016

In this paper we study first passage times of (reflected) Ornstein-Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry, Stadje and Zacks (2004) to the (reflected) Ornstein-Uhlenbeck case.
The Hartman-Watson Distribution Revisited: Asymptotics for Pricing Asian Options
Authors: Gerhold Stefan
Published: July 2016

Barrieu, Rouault and Yor (2004) determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution. We determine the asymptotics of the density. This refinement can be applied to the pricing of Asian options in the Black-Scholes model.
Proof of the Hamiltonicity-Trace Conjecture for Singularly Perturbed Markov Chains
Authors: Ejov Vladimir , Litvak Nelly , Nguyen Giang T. , Taylor Peter G.
Published: July 2016

We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles.
Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees
Authors: Munsonius Götz Olaf , Rüschendorf Ludger
Published: July 2016

Limit theorems are established for some functionals of the distances between two nodes in weighted randomb-ary recursive trees. We consider the depth of thenth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.
A Generalized Coupon Collector Problem
Authors: Xu Weiyu , Tang A. Kevin
Published: July 2016

This paper presents an analysis of a generalized version of the coupon collector problem, in which the collector receives d coupons each run and chooses the least-collected coupon so far. In the asymptotic case when the number of coupons n goes to infinity, we show that, on average, (nlogn) / d + (n / d)(m − 1)log logn + O(mn) runs are needed to collect m sets of coupons. An exact algorithm is also developed for any finite case to compute the exact mean number of runs. Numerical examples are provided to verify our theoretical predictions.
On Λ-Coalescents with Dust Component
Authors: Gnedin Alexander , Iksanov Alexander , Marynych Alexander
Published: July 2016

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a,b)-coalescents witha> 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.
Pricing path-dependent options in a Black-Scholes market from the distribution of homogeneous Brownian functionals
Authors: Fujita T. , Petit F. , Yor M.
Published: July 2016

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.
On the optimal allocation of service to impatient tasks
Authors: Glazebrook K. D. , Ansell P. S. , Dunn R. T. , Lumley R. R.
Published: July 2016

Service is often provided in contexts where tasks or customers are impatient or perishable in that they have natural lifetimes of availability for useful service. Moreover, these lifetimes are usually unknown to the service provider. The question of how service might best be allocated to the currently waiting tasks or customers in such a context has been neglected and we propose three simple models. For each model, an index heuristic is developed and is assessed numerically. In all cases studied the heuristic comes close to optimality.
The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment
Authors: Bérard Jean
Published: July 2016

The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabeiet al.for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for anarbitrarylevel of randomness.
Sequential selection of random vectors under a sum constraint
Authors: Stanke Mario
Published: July 2016

We observe a sequence X
1, X
2,…, X

n
of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X

i
we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]
d
and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {
x
∈ Q | 〈
x
,
θ
〉 ≤ 1}, where
θ
∈ ℝ
d
,
θ
≥ 0, satisfies a certain additional property, is a solution to this problem.
On-line parameter estimation for a failure-prone system subject to condition monitoring
Authors: Lin Daming , Makis Viliam
Published: July 2016

In this paper, we study the on-line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous-time homogeneous Markov process with a finite state space. The state of the system is hidden except for the failure state. When the system is operating, only the information obtained by condition monitoring, which is related to the working state of the system, is available. The condition monitoring observations are assumed to be in continuous range, so that no discretization is required. A recursive maximum likelihood (RML) algorithm is proposed for the on-line parameter estimation of the model. The new RML algorithm proposed in the paper is superior to other RML algorithms in the literature in that no projection is needed and no calculation of the gradient on the surface of the constraint manifolds is required. A numerical example is provided to illustrate the algorithm.
On the rate of Poisson process approximation to a Bernoulli process
Authors: Ruzankin P. S.
Published: July 2016

The main result of the paper is a refinement of Xia's (1997) bound on the Kantorovich distance between distributions of a Bernoulli point process and an approximating Poisson process. In particular, we show that the distance between distributions of a Bernoulli point process and the Poisson process with the same mean measure has the order of the total variation distance between the laws of the total masses of these processes.
The critical probability for the frog model is not a monotonic function of the graph
Authors: Fontes L. R. , Machado F. P. , Sarkar A.
Published: July 2016

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.
Distribution of the amount of genetic material from a chromosomal segment surviving to the following generation
Authors: Stefanov Valeri T.
Published: July 2016

A method is provided for numerical evaluation, with any given accuracy, of the probability that at leastp% of the genetic material from an individual's chromosomal segment survives to the next generation. Relevant MAPLE®V codes, for automated implementation of such evaluation, are also provided. The genomic continuum model, with Haldane's model for the crossover process, is assumed.
Critical growth of a semi-linear process
Authors: Molchanov Ilya , Shcherbakov Vadim , Zuyev Sergei
Published: July 2016

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X

t

− may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X

t

− means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.
A finite-time ruin probability formula for continuous claim severities
Authors: Ignatov Zvetan G. , Kaishev Vladimir K.
Published: July 2016

An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.
Spectral Factorization of Periodically Correlated MA(1) Processes
Authors: Bentarzi Mohamed , Hallin Marc
Published: July 2016

The spectral factorization problem, i.e. the problem of obtaining all possible MA representations of a process with given autocovariance function, is considered for univariate, d-periodic MA(1) (equivalently, 1-dependent in the second-order sense) processes. The solutions are provided explicitly, and their invertibility properties are investigated. A characterization, in terms of their autocovariance functions, of non-invertible d-periodic 1-dependent processes, extending to the periodic case the traditional unit root condition, is provided.
Increment-Vector Methodology: Transforming Non-Stationary Series to Stationary Series
Authors: Chen Zhao-Guo , Anderson Oliver D.
Published: July 2016

In time series analysis, it is well-known that the differencing operator ∇
d
may transform a non-stationary series, {Z(t)} say, to a stationary one, {W(t)} = ∇
d

Z(t)}; and there are many procedures for analysing and modelling {Z(t)} which exploit this transformation. Rather differently, Matheron (1973) introduced a set of measures on R

n
that transform an appropriate non-stationary spatial process to stationarity, and Cressie (1988) then suggested that specialized low-order analogues of these measures, called increment-vectors, be used in time series analysis. This paper develops a general theory of increment-vectors which provides a more powerful transformation tool than mere simple differencing. The methodology gives a handle on the second-moment structure and divergence behaviour of homogeneously non-stationary series which leads to many important applications such as determining the correct degree of differencing, forecasting and interpolation.
Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System
Authors: Guillemin Fabrice , Pinchon Didier
Published: July 2016

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ
t
} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ
t
}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.
Diffusion Approximation for Random Walks on Anisotropic Lattices
Authors: Horváth Lajos
Published: July 2016

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.
Optimal Stopping in a Dice Game
Authors: Roters Markus
Published: July 2016

In this paper we consider an explicit solution of an optimal stopping problem arising in connection with a dice game. An optimal stopping rule and the maximum expected reward in this problem can easily be computed by means of the distributions involved and the specific rules of the game
Large Deviations for Queue Lengths at a Multi-Buffered Resource
Authors: O'Connell Neil
Published: July 2016

In this paper we obtain the large deviation principle for scaled queue lengths at a multi-buffered resource, and simplify the corresponding variational problem in the case where the inputs are assumed to be independent.
On the variance to mean ratio for random variables from Markov chains and point processes
Authors: Brown Timothy C. , Hamza Kais , Xia Aihua
Published: July 2016

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.