Showing papers in "Stochastic Models in 2011"
TL;DR: In this article, the authors characterize the value function of maximizing the total discounted utility of dividend payments for a compound Poisson insurance risk model when strictly positive transaction costs are included, leading to an impulse control problem.
Abstract: We characterize the value function of maximizing the total discounted utility of dividend payments for a compound Poisson insurance risk model when strictly positive transaction costs are included, leading to an impulse control problem. We illustrate that well known simple strategies can be optimal in the case of exponential claim amounts. Finally we develop a numerical procedure to deal with general claim amount distributions.
33 citations
TL;DR: In this article, the authors considered discrete population genetics models with unequal (skewed) fertilities, with an emphasis on skewed versions of Cannings models, conditional branching process models in the spirit of Karlin and McGregor, and compound Poisson models.
Abstract: Discrete population genetics models with unequal (skewed) fertilities are considered, with an emphasis on skewed versions of Cannings models, conditional branching process models in the spirit of Karlin and McGregor, and compound Poisson models. Three particular classes of models with skewed fertilities are investigated, the Wright–Fisher model, the Dirichlet model, and the Kimura model. For each class the asymptotic behavior as the total population size N tends to infinity is investigated for power law fertilities and for geometric fertilities. This class of models can exhibit a rich variety of sub-linear or even constant effective population sizes. Therefore, the models are not necessarily in the domain of attraction of the Kingman coalescent. For a substantial range of the parameters, discrete-time coalescent processes with simultaneous multiple collisions arise in the limit.
31 citations
TL;DR: In this article, the authors proposed a method for detecting hidden regular variation, and when it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure.
Abstract: Hidden regular variation requires regular variation on 𝔼 = [0, ∞] d \ {(0, 0,…, 0)} and another regular variation on the sub-cone , where 𝕃 i is the ith axis. We extend this concept to sub-cones of 𝔼(2) as well. We suggest a procedure for detecting hidden regular variation, and when it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure. We give an example where hidden regular variation yields improved estimates of probabilities of risk sets.
28 citations
TL;DR: In this paper, the authors consider dynamic admission and termination control policies in a Markovian loss system with two classes, each with a fixed reward, a termination cost, an arrival and service rate.
Abstract: We consider dynamic admission and termination control policies in a Markovian loss system with two classes, each with a fixed reward, a termination cost, an arrival and service rate. The system may admit or reject an arriving job or admit it by terminating a job in the system to maximize its total expected discounted reward. We prove that (1) when there is an idle server, it is never optimal to terminate a job, (2) there exists an optimal threshold policy for both admission and termination decisions. Furthermore, we identify the conditions which ensure that a class is “preferred” or “strongly preferred.”
24 citations
TL;DR: In this paper, the authors investigated the statistical properties of wave characteristics related to wave asymmetry in the 3D Lagrange model and proposed a modification of the original Lagrangian model that can produce front-back asymmetry both of the space waves and of the time waves, observed at a fixed measuring station.
Abstract: In the stochastic Lagrange model for ocean waves the vertical and horizontal location of surface water particles are modeled as correlated Gaussian processes. In this article we investigate the statistical properties of wave characteristics related to wave asymmetry in the 3D Lagrange model. We present a modification of the original Lagrange model that can produce front-back asymmetry both of the space waves, i.e. observation of the sea surface at a fixed time, and of the time waves, observed at a fixed measuring station. The results, which are based on a multivariate form of Rice's formula for the expected number of level crossings, are given in the form of the cumulative distribution functions for the slopes observed either by asynchronous sampling in space, or at synchronous sampling at upcrossings and down-crossings, respectively, of a specified fixed level. The theory is illustrated in a numerical section, showing how the degree of wave asymmetry depends on the directional spectral spreading and on t...
20 citations
TL;DR: In this paper, a discrete time Markov chain for the accumulated weight of orders in the system is introduced and analyzed, and the distributions of the accumulated weights at an arbitrary time, total accumulated weights in a consolidation cycle, and excess of weight per shipment are obtained.
Abstract: This article studies the dispatch of consolidated shipments. Orders arrive to a depot at discrete time epochs following a discrete time batch Markov arrival process (BMAP). The weight of an order is measured in discrete units and may be correlated with the arrival time. As soon as the total weight of the accumulated orders reaches a threshold, which is a function of the time elapsed since the last dispatch, all orders are consolidated and a shipment is dispatched. A discrete time Markov chain for the accumulated weight of orders in the system is introduced and analyzed. The distributions of the accumulated weight at an arbitrary time, total accumulated weight in a consolidation cycle, and excess of weight per shipment are obtained. By introducing an absorption Markov chain and a terminating Markovian arrival process, we find the distributions of the consolidation cycle length, the waiting time of an arbitrary order, and the number of orders that occur in a cycle. An efficient computational procedure is de...
18 citations
TL;DR: In this article, the authors study the waiting-time distributions in cyclic polling models with renewal arrivals, general service and switch-over times, and exhaustive service at each of the queues.
Abstract: We study the waiting-time distributions in cyclic polling models with renewal arrivals, general service and switch-over times, and exhaustive service at each of the queues. The assumption of renewal arrivals prohibits an exact analysis and reduces the available analytic results to heavy-traffic asymptotics, limiting results for large switch-over times and large numbers of queues, and some numerical algorithms. Motivated by this, the goal of this paper is to propose a new method for deriving simple closed-form approximations for the complete waiting-time distributions that work well for arbitrary load values. Extensive simulation results show that the approximations are highly accurate over a wide range of parameter settings.
17 citations
TL;DR: A spatial stochastic model for telecommunication networks, the stoChastic subscriber line model, is considered, and the distribution of the typical shortest path length between network components is investigated, and a representation formula for the probability density is derived based on functionals of the so-called typical serving zone.
Abstract: We consider a spatial stochastic model for telecommunication networks, the stochastic subscriber line model, and we investigate the distribution of the typical shortest path length between network components. Therefore, we derive a representation formula for the probability density of this distribution which is based on functionals of the so-called typical serving zone. Using this formula, we construct an estimator for the density of the typical shortest path length and we analyze the statistical properties of this estimator. Moreover, we introduce new simulation algorithms for the typical serving zone which are used in a numerical study in order to estimate the density and moments of the typical shortest path length for different specific models.
14 citations
TL;DR: In this paper, the authors discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices, and show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations.
Abstract: Mandatory emission trading schemes are being established around the world. Participants of such market schemes are always exposed to risks. This leads to the creation of an accompanying market for emission-linked derivatives. To evaluate the fair prices of such financial products, one needs appropriate models for the evolution of the underlying assets, emission allowance certificates. In this paper, we discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices. We show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations. The results are illustrated by a small numerical study.
14 citations
TL;DR: In this article, the authors studied the local times of a Levy process reflected at two barriers 0 and K > 0 and derived asymptotics for l K when the Levy process has mean zero.
Abstract: The present paper is concerned with the local times of a Levy process reflected at two barriers 0 and K > 0. The reflected process is decomposed into the original process plus local times at 0 and K and a starting condition, and we study l K , the mean rate of increase of the local time at K when the reflected process is started in stationarity. We derive asymptotics (K → ∞) for l K when the Levy process has mean zero. The precise form of the asymptotics depends on the existence or non-existence of a finite second moment, paralleling the difference between the normal and the stable central limit theorem.
13 citations
TL;DR: In this paper, the authors present an optimization approach to weak approximation of a general class of stochastic differential equations with jumps, in particular when value functions with compact support are considered.
Abstract: We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach employs a mathematical programming technique yielding upper and lower bounds of the expectation, without Monte Carlo sample paths simulations, based upon the exponential tempering of bounding polynomial functions to avoid their explosion at infinity. The resulting tempered polynomial optimization problems can be transformed into a solvable polynomial programming after a minor approximation. The exponential tempering widens the class of stochastic differential equations for which our methodology is well defined. The analysis is supported by numerical results on the tail probability of a stable subordinator and the survival probability of Ornstein-Uhlenbeck processes driven by a stable subordinator, both of which can be formulated with value functions with compact support and are not applicable ...
TL;DR: In this article, the authors provide a probabilistic phase-type representation for the first order moment distributions and an alternative representation, with an analytically appealing form, for the latter.
Abstract: Moment distributions of phase-type and matrix-exponential distributions are shown to remain within their respective classes. We provide a probabilistic phase-type representation for the former case and an alternative representation, with an analytically appealing form, for the latter. First order moment distributions are of special interest in areas like demography and economics, and we calculate explicit formulas for the Lorenz curve and Gini index used in these disciplines.
TL;DR: In this article, the authors consider two insurance companies that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal).
Abstract: Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This article is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying ...
TL;DR: In this paper, the authors considered a risk process which exbihits the key features of companies with steady outflows and sporadic inflows (e.g., discoveries, patents), and a risk management policy was further implemented stating that the outflow rate is reduced when no revenue (inflow) is generated within an Erlang n time period.
Abstract: In this article, we consider a risk process which exbihits the key features of companies with steady outflows and sporadic inflows (e.g., discoveries, patents). A risk management policy is further implemented stating that the outflow rate is reduced when no revenue (inflow) is generated within an Erlang-n time period. For the surplus process of interest, a Markovian representation is first given which leads to the form of the solution for the Laplace transform of the time to ruin. A homogeneous linear integro-differential equation for the Laplace transform of the time of ruin is later derived. The boundary conditions of the aforementioned integro-differential equation are used to complete the representation of the Laplace transform of the time to ruin. Finally, numerical applications are considered to illustrate the effectiveness of this risk management policy to lower the company's solvency risk.
TL;DR: In this article, the authors generalized the class of risk models with Markovian arrival process (MAP) by allowing the waiting times between two successive events (which can be a change in the environmental state and/or a claim arrival) to have an arbitrary distribution.
Abstract: The class of risk models with Markovian arrival process (MAP) (see e.g., Neuts[ 15 ]) is generalized by allowing the waiting times between two successive events (which can be a change in the environmental state and/or a claim arrival) to have an arbitrary distribution. Using a probabilistic approach, we determine the solution for a class of Gerber–Shiu functions apart from some unknown constants when claim sizes have a mixed exponential distribution. Such constants are later determined using the more classic ruin-analytic approach. A numerical example is later considered to illustrate the tractability of the suggested methodology in the study of Gerber–Shiu functions.
TL;DR: This work aims to scrutinise Chargaff's second parity rule by developing tests of statistical significance for the rule and then applying them to a large set of bacterial genomes taken from the GenBank repository.
Abstract: Chargaff's first parity rule is a property of double-stranded DNA which states that the number of A and T nucleotides, and the number of C and G nucleotides, are the same within the duplex. It arises as a result of the chemistry of nucleic acids which only permits A to bond with T and C to bond with G. In contrast, Chargaff's second parity rule asserts that the same is also true within a single strand, not only for mononucleotide chains, but also for short polynucleotide chains. Unlike the first parity rule, the second is not exact[ 12 ]. Several explanations for the origins of this intrastrand symmetry have been proposed, but the relative contribution of these mechanisms to the symmetry are still not clear[ 19 ]. This work aims to scrutinise Chargaff's second parity rule by developing tests of statistical significance for the rule and then applying them to a large set of bacterial genomes taken from the GenBank repository. We also consider the vector of mononucleotide frequencies (π a : a ∈ {A, C, G, T})...
TL;DR: In this article, a multidimensional insurance model is proposed in terms of Skorokhod problem in an orthant, to describe the dynamics of d companies operating under a risk-reducing treaty.
Abstract: A multidimensional insurance model is proposed in terms of Skorokhod problem in an orthant, to describe the dynamics of d companies operating under a risk-reducing treaty. Investment in risky assets and fluctuations can also be incorporated into the model. No assumptions on moments of claims are needed. The coefficients can depend on the ‘pushing part’ also. Wellposedness of the model is established. In the case of some Markovian examples, the infinitesimal generators are identified; an interesting aspect is the appearance of the linear complementarity problem in the generator.
TL;DR: An enhanced version of the splitting method, called the smoothed splitting method (SSM), for counting associated with complex sets, such as the set defined by the constraints of an integer program and in particular for counting the number of satisfiability assignments.
Abstract: We present an enhanced version of the splitting method, called the smoothed splitting method (SSM), for counting associated with complex sets, such as the set defined by the constraints of an integer program and in particular for counting the number of satisfiability assignments. Like the conventional splitting algorithms, ours uses a sequential sampling plan to decompose a "difficult" problem into a sequence of "easy" ones. The main difference between SSM and splitting is that it works with an auxiliary sequence of continuous sets instead of the original discrete ones. The rationale of doing so is that continuous sets are easier to handle. We show that while the proposed method and its standard splitting counterpart are similar in their CPU time and variability, the former is more robust and more flexible than the latter.
TL;DR: In this paper, the authors discuss ruin theory when the insurance risk process is described by a hidden Markov, regime-switching diffusion process, and derive a partial differential equation for the ruin probability in the filtered model.
Abstract: We discuss ruin theory when the insurance risk process is described by a hidden Markov, regime-switching diffusion process. The innovations approach to filtering theory is used to transform the partially observed modeling framework into one with complete observations. (Robust) filters for the hidden states of the chain are given. A partial differential equation for the ruin probability is derived in the “filtered” model.
TL;DR: In this paper, the authors use semi-regenerative, fluctuations methods, and sequential games to examine a queueing system with server performing a secondary maintenance work, and obtain compact and explicit functionals for the queueing process in equilibrium for both discrete and continuous time parameter processes.
Abstract: We use semi-regenerative, fluctuations methods, and sequential games to examine a queueing system with server performing a secondary maintenance work. When the buffer becomes empty, the server leaves the system to service packets of jobs at a maintenance facility where his time there is restricted by T (a random time). He returns to the system when his sojourn time expires and if the last packet is fully processed. If the buffer contents is below N, the server departs from the system again to continue with maintenance; this time rendering just one packet at a time until the buffer is replenished. We obtain compact and explicit functionals for the queueing process in equilibrium for both discrete and continuous time parameter processes.
TL;DR: In this article, the authors show that the generalized Pareto family does not share the linearity of the mean excess plot, which is consistent with the underlying distribution of thresholded data.
Abstract: In risk analysis, the mean excess plot is a commonly used exploratory plotting technique for confirming iid data is consistent with a generalized Pareto assumption for the underlying distribution since in the presence of such a distribution, thresholded data have a mean excess plot that is roughly linear. Does any other class of distributions share this linearity of the plot? Under some extra assumptions, we are able to conclude that only the generalized Pareto family has this property.
TL;DR: In this article, a stable skip free reflected random walk in the two dimensional nonnegative quadrant, where the main state is referred to as level, is considered and the authors establish the asymptotic distribution of the second coordinate (phase) at the moment the process hits a large level for the first time.
Abstract: Consider a stable skip free reflected random walk in the two dimensional nonnegative quadrant, where the main state is referred to as level. We establish the asymptotic distribution of the second coordinate (phase) at the moment the process hits a large level for the first time. The limit is derived under two scenarios. In the first, the chain reaches the destination level for the first time before returning to the starting level and in the second it reaches the destination level for the first time before returning to the starting state. We will show that under the same conditions the limits are equal. The existence of the hitting distribution in limit is surprising in the bridge case where the large deviations path tries to avoid the level axis. We also show that the reciprocal of the mean time to reach a level for the first time decays at a rate which is the same as the decay rate of the stationary distribution. Moreover, the results are applied to a modified Jackson network with partially coupled servers.
TL;DR: The results extend those of existing models by showing that two possible regimes of long-range dependence exist at different time scales and the pertinence of the proposed model is validated on real web traffic traces, and its ability to accurately explain the Hurst parameter is validate on both web traces and numerical simulations.
Abstract: For more than a decade, it has been observed that network traffic exhibits long-range dependence and many models have been proposed relating this property to heavy-tailed flow durations. However, none of these models consider correlations at flow scale. Such correlations exist and will become more prominent in the future Internet with the emergence of flow-aware control mechanisms correlating a flow's transmission to its characteristics (size, duration, etc.). In this article, we study the impact of the correlation between flow rates and durations on the long-range dependence of aggregate traffic. Our results extend those of existing models by showing that two possible regimes of long-range dependence exist at different time scales. The long-range dependence in each regime can be stronger or weaker than standard predictions, depending on the conditional statistics between the flow rates and durations. In the independent case, our proposed model consistently reduces to former approaches. The pertinence of ...
TL;DR: In this paper, the authors studied the extinction probability of Markovian binary trees subject at random epochs to catastrophes, controlled by a Markov arrival process, and killing random numbers of living individuals.
Abstract: We study transient features and the extinction probability of a particular class of multi-type Markovian branching processes called Markovian binary trees, subject at random epochs to catastrophes, controlled by a Markovian arrival process, and killing random numbers of living individuals. We provide two types of probabilistic methods to numerically compute the extinction probability: the first one is based on an integral equation for the probability generating function of the population size, and the second one is based on the correspondence between Markovian branching processes with catastrophes and structured Markov chains of G/M/1-type.
TL;DR: In this article, the structural properties of the moments of Markov arrival processes (MAPs) and Rational Arrival Process (RAPs) are investigated and it is shown that redundant RAPs/MAPs of order n are characterized by less than n 2 independent moments.
Abstract: The structural properties of the moments of Markov arrival processes (MAPs) and Rational arrival processes (RAPs) are considered in this article. We investigate how many and which moments can characterize these processes and show that redundant RAPs/MAPs of order n are characterized by less than n 2 independent moments.
TL;DR: In this paper, the invariant measure of a generalized Markov branching process with non-boundary transitions is investigated. But the invariance measure is not invariant in the case of a truncated model.
Abstract: In this paper, we obtain the asymptotic properties of the invariant measure of a generalized Markov branching process according to three different cases of non-boundary transitions. These results are applied to investigate the asymptotic behavior of the corresponding truncated model.
TL;DR: In this article, the authors considered the M/G/∞ model, where jobs arrive according to a Poisson process with rate λ, and each of them stays in the system during a random amount of time, distributed as a non-negative random variable B; throughout it is assumed that B is light-tailed.
Abstract: This paper considers the so-called M/G/∞ model: jobs arrive according to a Poisson process with rate λ, and each of them stays in the system during a random amount of time, distributed as a non-negative random variable B; throughout it is assumed that B is light-tailed. With N(t) denoting the number of jobs in the system, the random process A(t) records the load imposed on the system in [0, t], i.e., A(t):= ∫t0 N(s)ds. The main result concerns the tail asymptotics of A(t)/t: we find an explicit function f(·) such that f(t) ∼ IP(A(t)/t > ρ(1+e). for t large; here ρ: =λ double struck E sign B. A crucial issue is that A(t) does not have i.i.d. increments, which makes direct application of the classical Bahadur-Rao result impossible; instead an adaptation of this result is required. We compare the asymptotics found with the (known) asymptotics for ρ → ∞ (and t fixed). Copyright © Taylor & Francis Group, LLC.
TL;DR: In this article, the M-matrix technique was used to perturb the functional Kolmogorov-type system with infinite delay into the infinite delay stochastic functional differential system.
Abstract: In general, time delay and system uncertainty are commonly encountered for all population systems. To examine whether the presence of environmental noise affects infinite delay population systems significantly, this paper perturbs the functional Kolmogorov-type system with infinite delay into the infinite delay stochastic functional differential system By the M-matrix technique, this paper examines the global positive solution and its pathwise estimation for this stochastic functional Kolmogorov-type population system. To illustrate the applications of our theory more clearly, this paper also discusses a Lotka–Volterra system with mixed delays as a special case.
TL;DR: In this article, the authors approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter, by considering two kinds of warranty policies, i.e., the non-renewing free replacement warranty policy and the non renewing pro-rata warranty policy.
Abstract: We approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter. We consider two kinds of warranty policies, namely, the non-renewing free replacement warranty policy and the non-renewing pro-rata warranty policy. Our approximation holds under modest assumptions on the distribution of the sales process of the warranted item and the nature of arrivals of warranty claims. We propose a method of using historical data to statistically estimate the parameters of the approximate distribution. Our methodology is applied to the warranty claims data from a large car manufacturer for a single car model and model year.