Stochastic Models

Yuan, Haili; Hu, Yijun

doi: 10.1080/15326340902869846pmid: N/A

We consider the compound Poisson risk model with a constant force of interest and a threshold strategy. Under such a strategy, no dividends are paid if the insurer's surplus is below a certain threshold level. When the surplus is above the threshold level, part of the premium income and all of the interest income are paid out as dividends. The integro-differential equations for the Gerber–Shiu discounted penalty function and the expected discounted dividends are derived and solved. Closed-form expressions are given when the claim size is exponentially distributed. Numerical presentations are also provided for the case of exponential individual claim to illustrate the influence of force of interest and the safety loading on the expected discounted dividends.

Es-Saghouani, Abdelghafour; Mandjes, Michel

doi: 10.1080/15326340902869887pmid: N/A

In this article we study Gaussian queues (that is, queues fed by Gaussian processes, such as fractional Brownian motion (fBm) and the integrated Ornstein–Uhlenbeck (iOU) process), with a focus on the dependence structure of the workload process. The main question is to what extent does the workload process inherit the dependence properties of the input process? We first present a specific notion of dependence that allows (in asymptotic regimes) explicit analysis. For the special cases of fBm and iOU, we analyze the behavior of this metric under a many sources scaling. Relying on (the generalized version of) Schilder's theorem, we are able to characterize its decay. We observe that the dependence structure of the input process essentially carries over to the workload process (in the asymptotic regime that we have chosen, in terms of our specific notion of dependence).

Dimitriou, Ioannis; Langaris, Christos

doi: 10.1080/15326340902869945pmid: N/A

We consider a queueing system with Poisson arrivals and arbitrarily distributed service times, vacation times, and start-up and close-down times. The model accepts two types of customers—the ordinary and the retrial customers—and the server takes a single vacation each time he becomes free. For such a model the stability conditions and the system state probabilities are investigated both in a transient and in the steady state. Numerical results are finally obtained and used to observe system performance for various values of the parameters.

Kadankov, V.; Kadankova, T.; Veraverbeke, N.

doi: 10.1080/15326340902869978pmid: N/A

In this article we determine the Laplace transforms of the one-boundary characteristics and the distribution of the number of intersections of a fixed interval by a difference of a compound Poisson process and a compound renewal process. The results obtained are applied for a particular case of this process, namely, for the difference of the compound Poisson process and the renewal process whose jumps are geometrically distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. In this case, under certain assumptions, we find the limit distributions of the one-boundary and two-boundary characteristics of the process. In addition, we prove the weak convergence of these distributions to the corresponding distributions of a symmetric Wiener process.

Michel, Julien; Porret-Blanc, Sylvain

doi: 10.1080/15326340902870091pmid: N/A

Distribution networks in random environments provide many models in stochastic geometry. The one addressed here concerns the case of a road network described by the classical homogenous Poisson line process[ 17 , 18 ,25] the clients are located in the plane, and are connected to the nearest road segment. The aim of this article is to show the existence of empirical and typical moments of the main geometric characteristics of the cells of the induced tessellation. A numerical study of the typical cell thanks to the realization of the tessellation via a Palm representation is given, showing the difference between this typical cell and the classical Crofton cell. This simulation also gives the correlation between two adjacent cells of the Poisson line tessellation.

Zou, Jiezhong; Zhang, Zhenzhong; Zhang, Jiankang

doi: 10.1080/15326340902870133pmid: N/A

This article considers the dividend optimization problem for an insurer with a jump-diffusion risk process in the presence of fixed and proportional transaction costs. Due to the presence of a fixed transaction cost, the mathematical problem becomes an impulse stochastic control problem. Using a stochastic impulse control approach, we transform the stochastic control problem into a quasi-variational inequality for a second-order nonlinear integro-differential equation. Under a risk-neutral assumption for the insurer, we solve this problem explicitly and construct the value function together with the optimal policy. Finally, we discuss the expected time to the first dividend payment when the optimal strategy is employed.

Rahimov, I.

doi: 10.1080/15326340902870158pmid: N/A

We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.