Showing papers in "Stochastic Models in 2014"

Bayesian Dividend Optimization and Finite Time Ruin Probabilities

Abstract: We consider the valuation problem of an (insurance) company under partial information. Therefore we use the concept of maximizing discounted future dividend payments. The firm value process is described by a diffusion model with constant and observable volatility and constant but unknown drift parameter. For transforming the problem to a problem with complete information, we derive a suitable filter. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We state a numerical procedure for approximating both the optimal dividend strategy and the corresponding value function. Furthermore, threshold strategies are discussed in some detail. Finally, we calculate the probability of ruin in the uncontrolled and controlled situation.

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

Abstract: Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N2, respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N2 distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an o...

Queues and Risk Processes with Dependencies

Abstract: We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in Ref.[13]. In this setting, we obtain the steady-state waiting time distribution, and we show that the classical relation between the steady-state waiting time and workload distributions remains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally, we show that there exist stochastic order relations between the waiting times under various instances of correlation.

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Abstract: Let {Xi(t), t ⩾ 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper, we derive the exact asymptotics of the joint survival function as u → ∞. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X1, X2. As a by-product, we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields.

Lévy Processes, Phase-Type Distributions, and Martingales

Abstract: Levy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Levy processes (tempered stable processes), NIG (normal inverse Gaussian) Levy processes, and hyperbolic Levy processes. We consider here a dense class of Levy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Levy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Levy process with reflection in 0. The solutions are in terms of roots to polynomi...

On the Evolution and Asymptotic Analysis of Open Markov Populations: Application to Consumption Credit

Abstract: In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that state probabilities within transient states converge—even when the overall expected population dimension increases without bound—under general conditions on the transition matrix and input intensities.Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow for the forecast, by confidence intervals, of the evolution of the relative population structure in the transient states.Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.

A Tree-Structured Markovian Model of the Shipment Consolidation Process

Abstract: This article studies the dispatch of consolidated shipments. Orders, following a batch Markovian arrival process, are received in discrete quantities by a depot at discrete time epochs. Instead of immediate dispatch, all outstanding orders are consolidated and shipped together at a later time. The decision of when to send out the consolidated shipment is made based on a “dispatch policy,” which is a function of the system state and/or the costs associated with that state. First, a tree structured Markov chain is constructed to record specific information about the consolidation process; the effectiveness of any dispatch policy can then be assessed by a set of long-run performance measures. Next, the effect on shipment consolidation of varying the order-arrival process is demonstrated through numerical examples and proved mathematically under some conditions. Finally, a heuristic algorithm is developed to determine a favorable parameter of a special set of dispatch policies, and the algorithm is proved to ...

First passage times to congested states of many-server systems in the Halfin-Whitt regime

Abstract: We consider the heavy-traffic approximation to the GI/M/s queueing system in the Halfin–Whitt regime, where both the number of servers s and the arrival rate λ grow large (taking the service rate as unity), with and β some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves as a Brownian motion with drift above zero and as an Ornstein–Uhlenbeck process below zero. We analyze the first passage times of this hybrid diffusion process to levels in the state space that represent congested states in the original queueing system.

On Transition and First Hitting Time Densities and Moments of the Ornstein–Uhlenbeck Process

Abstract: This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions

The topology of central counterparty clearing networks and network stability

Abstract: □ This paper derives a measure of central counterparty (CCP) clearing-network risk that is based on the probability that the maximum exposure (the N-th order statistic) of a CCP to an individual general clearing member is large. Our analytical derivation of this probability uses the theory of Laplace asymptotics, which is related to the large deviations theory of rare events. The theory of Laplace asymptotics is an area of applied probability that studies the exponential decay rate of certain probabilities and is often used in the analysis of the tails of probability distributions. We show that the maximum-exposure probability depends on the topology, or structure, of the clearing network. We also derive a CCP's Maximum-Exposure-at-Risk, which provides a metric for evaluating the adequacy of the CCP's and general clearing members’ loss-absorbing financial resources during rare but plausible market conditions. Based on our analysis, we provide insight into how clearing-network structure can affect the maxi...

Gambler’s Ruin: The Duration of Play

Abstract: We study the gambler’s ruin problem with a general distribution of the payoffs in each game. Assuming the expected value of the payoff distribution is negative, so that eventual ruin occurs with probability 1, we are interested in the distribution of the duration to ruin, also known as the first-passage time distribution. A generating function for this distribution is obtained. Exact expressions for the expected value and variance of this distribution, as well as asymptotic expressions for the case of large initial wealth, are derived.

A Nonstationary Cylinder–Based Model Describing Group Dispersal in a Fragmented Habitat

Abstract: □ A doubly nonstationary cylinder-based model is built to describe the dispersal of a population from a point source. In this model, each cylinder represents a fraction of the population, i.e., a group. Two contexts are considered: The dispersal can occur in a uniform habitat or in a fragmented habitat described by a conditional Boolean model. After the construction of the models, we investigate their properties: the first and second order moments, the probability that the population vanishes, and the distribution of the spatial extent of the population.

Blocking probabilities in Erlang loss queues with advance reservation

Abstract: We study the blocking probability in a continuous time loss queue, in which resources can be claimed a random time in advance. We identify classes of loss queues where the advance reservation results in increased or decreased blocking probabilities. The lower blocking probabilities are achieved because the system tends to favor short jobs. We provide analytical and numerical results to establish the connection between the system’s parameters and either an increase or decrease of blocking probabilities, compared to the system without reservation.

Signatures in Markov-Modulated Processes

Abstract: □ In recent years, signatures are widely used for analysis of coherent systems consisting of unreliable components. If component lifetimes are independent and identically distributed, then system lifetime distribution function is a convex combination of distribution functions of order statistics for component lifetimes. Coefficients of this convex combination are called signatures. This article considers the case when a system operates in a so-called random environment, i.e., component failure rates are jointly modulated by a finite-state continuous-time Markov chain. In this model, component lifetimes remain exchangeable. An expression for distribution function of time to system failure is derived. Here, a crucial role is played by an elaborated procedure of deriving a distribution function of order statistics for system component lifetimes. A numerical example illustrates the suggested approach and analyzes the influence of random environment on the distribution function of system lifetime.

Phase Type Distributions with Finite Support

Abstract: This research is motivated by the fact that many random variables of practical interest have a finite support. For fixed a < b, we consider the distribution of a random variable X = (a + Ymod(b − a)), where Y is a phase type (PH) random variable. We demonstrate that as we traverse for Y the entire set of PH distributions (or even any subset thereof like Coxian that is dense in the class of distributions on [0, ∞)), we obtain a class of matrix exponential distributions dense in (a, b). We call these Finite Support Phase Type Distributions (FSPH) of the first kind. A simple example shows that though dense, this class by itself is not very efficient for modeling; therefore, we introduce (and derive the EM algorithms for) two other classes of finite support phase type distributions (FSPH). The properties of denseness, connection to Markov chains, the EM algorithm, and ability to exploit matrix-based computations should all make these classes of distributions attractive not only for applied probability but als...

Uniform Tail Asymptotics for the Sum of Two Correlated Classes with Stochastic Returns and Dependent Heavy Tails

Abstract: Consider a continuous-time risk model with two correlated classes of insurance business and risky investments whose price processes are geometric Levy processes. By assuming that the correlation comes from a common shock, and the claim sizes are heavy-tailed and pairwise quasi-asymptotically independent, we investigate the tail behavior of the sum of the stochastic present values of the two correlated classes, and a uniform asymptotic formula is obtained.

Optimal Strategies for a Long-Term Static Investor

Abstract: The optimal strategies for a long-term static investor are studied. Given a portfolio of a stock and a bond, we derive the optimal allocation of the capitals to maximize the expected long-term growth rate of a utility function of the wealth. When the bond has a constant interest rate, three models for the underlying stock price processes are studied: Heston model, 3/2 model, and jump diffusion model. We also study the optimal strategies for a portfolio in which the stock price process follows a Black-Scholes model and the bond process has a Vasicek interest rate that is correlated to the stock price.

Performance Analysis of Exponential Multi-Server Production Lines with Fluid Flow and Finite Buffers

Abstract: This article presents an approximation method for fluid flow production lines with multi-server workstations and finite buffers. Each workstation consists of parallel identical servers, which are subject to operation-dependent failures with exponentially distributed uptimes and downtimes. The proposed method decomposes the production line into single-buffer subsystems, each described by a continuous state Markov process, the parameters of which are determined iteratively. The approximation method is appropriate for the analysis of longer production lines, able to accurately estimate performance characteristics (e.g., throughput and mean buffer content), and shown to perform well on a large test set.

Regular Variation of Infinite Series of Processes with Random Coefficients

Abstract: In this article, we consider a series X(t) = ∑j ⩾ 1Ψj(t)Zj(t), t ∈ [0, 1] of random processes with sample paths in the space of cadlag functions (i.e., right-continuous functions with left limits) on [0, 1]. We assume that (Zj)j ⩾ 1 are i.i.d. processes with sample paths in , and (Ψj)j ⩾ 1 are processes with continuous sample paths. Using the notion of regular variation for -valued random elements (introduced in Ref.[13]), we show that X is regularly varying if Z1 is regularly varying, (Ψj)j ⩾ 1 satisfy some moment conditions, and a certain “predictability assumption” holds for the sequence {(Zj, Ψj)}j ⩾ 1. Our result can be viewed as an extension of Theorem 3.1 of Ref.[15] from random vectors in to random elements in . As a preliminary result, we prove a version of Breiman’s lemma for -valued random elements, which can be of independent interest.

The Maximum of Independent Geometric Random Variables as the Time for Genomic Evolution

Abstract: □ We calculate the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, geometric random variables. In many cases, the limit distribution turns out to be the standard Gumbel. The motivation comes from a variant of the genomic evolutionary model proposed by Wilf and Ewens[ 15 ] as an answer to the criticism of the Darwinian theory of evolution stating that the time required for the appropriate mutations is huge. A byproduct of our analysis is the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, exponential random variables.

Stochastic Bounds with a Low Rank Decomposition

Abstract: We investigate how we can bound a discrete time Markov chain (DTMC) by a stochastic matrix with a low rank decomposition. In the first part of the article, we show the links with previous results for matrices with a decomposition of size 1 or 2. Then we show how the complexity of the analysis for steady-state and transient distributions can be simplified when we take into account the decomposition. Finally, we show how we can obtain a monotone stochastic upper bound with a low rank decomposition.

Critical Scale for a Continuous AIMD Model

Abstract: A scaled version of the general AIMD model of transmission control protocol (TCP) used in Internet traffic congestion management leads to a Markov process x(t) representing the time dependent data flow that moves forward with constant speed on the positive axis and jumps backward to γx(t), 0 < γ < 1 according to a Poisson clock whose rate α(x) depends on the interval swept in between jumps. We give sharp conditions for Harris recurrence and analyze the convergence to equilibrium on multiple scales (polynomial, fractional exponential, exponential) identifying the critical case xα(x) ∼ β. Criticality has different behavior according to whether it occurs at the origin or infinity. In each case, we determine the transient (possibly explosive), null—and positive—recurrent regimes by comparing β to ( − ln γ)− 1.

On the Asymptotic Behavior of a Dynamic Version of the Neyman Contagious Point Process

Abstract: We consider a dynamic version of the Neyman contagious point process that can be used for modeling the spacial dynamics of biological populations, including species invasion scenarios. Starting with an arbitrary finite initial configuration of points in with nonnegative weights, at each time step a point is chosen at random from the process according to the distribution with probabilities proportional to the points’ weights. Then a finite random number of new points is added to the process, each displaced from the location of the chosen “mother” point by a random vector and assigned a random weight. Under broad conditions on the sequences of the numbers of newly added points, their weights and displacement vectors (which include a random environments setup), we derive the asymptotic behavior of the locations of the points added to the process at time step n and also that of the scaled mean measure of the point process after time step n → ∞.

The Lamperti Transforms of Self-Similar Gaussian Processes and Their Exponentials

Abstract: We present results on the second order behavior and the expected maximal increments of Lamperti transforms of self-similar Gaussian processes and their exponentials. The Ornstein Uhlenbeck processes driven by fractional Brownian motion (fBM) and its exponentials have been recently studied in Ref.[ 20 ] and Ref.[ 21 ], where we essentially make use of some particular properties, e.g., stationary increments of fBM. Here, the treated processes are fBM, bi-fBM, and sub-fBM; the latter two are not of stationary increments. We utilize decompositions of self-similar Gaussian processes and effectively evaluate the maxima and correlations of each decomposed process. We also present discussion on the usage of the exponential stationary processes for stochastic modeling.

Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment

Abstract: We develop accurate approximations for the delay distribution of the MArP/G/1 queue that capture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in a MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.

The discounted moments of the surplus after the last innovation before ruin under the dual risk model

Abstract: □ This article's focus is on finding an explicit form of the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. For this purpose, we derive a non-homogeneous integro-differential equation, which is satisfied by the targeted quantity. To solve this equation, the general solution of the corresponding homogeneous equation and a particular solution of the non-homogeneous equation are obtained. Also, some additional results are provided, such as the defective distribution of the time to ruin and the Laplace transform of the time when the last jump before ruin happens.