Stochastic Models

Basrak, Bojan; Kulik, Rafał; Palmowski, Zbigniew

doi: 10.1080/15326349.2013.838508pmid: N/A

In this article, we analyze a branching process with immigration defined recursively by X t  = θ t ○ X t−1 + B t for a sequence (B t ) of i.i.d. random variables and random mappings , with being a sequence of ℕ0-valued i.i.d. random variables independent of B t . We assume that one of generic variables A and B has a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solution X t . We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.

Ivanov, Roman V.

doi: 10.1080/15326349.2013.838509pmid: N/A

In this article, the author assumes that the stock price dynamics follows the exponential of the normal-inverse Gaussian process. Analytical formulas for values of digital options and European calls are obtained. The considered family of the four-parametric normal-inverse Gaussian processes has steepness parameter . The established formulas depend on the values of the degenerate Appell hypergeometric function.

Shang, Yilun

doi: 10.1080/15326349.2013.838510pmid: N/A

The geometry of complex networks has a close relationship with their structure and function. In this article, we investigate Gromov-hyperbolicity of inhomogeneous random networks modeled by the Chung-Lu model G(w). When the maximum expected degree w max and minimum expected degree w min satisfy w max ≤ 21/3 w min, we prove that for any positive δ, G(w) has a positive probability of containing δ-fat triangles as n → ∞. Our numerical simulations illustrate this non-hyperbolicity of G(w) for power law degree distributions among others.

Blom, Joke; De Turck, Koen; Mandjes, Michel

doi: 10.1080/15326349.2013.838511pmid: N/A

This article studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates λ i by a factor N and the rates ν ij of the background process by N 1+ϵ (for some ϵ > 0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that N tends to ∞. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean.

Muratov, Anton; Zuyev, Sergei

doi: 10.1080/15326349.2013.839191pmid: N/A

We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This class covers, in particular, generalized Polya urn scheme, Dubins–Freedman random measures, and cooperative sequential adsorption models studied previously. Specifically, we address models where the distribution of a newly added particle is determined by the distance to the closest particle from the chosen one. We address boundedness of the processes and convergence properties of the corresponding sample measure. We show that, in general, the limiting measure is random when it exists and that this is the case for a wide class of almost surely bounded processes.

Ghosh, Arka P.

doi: 10.1080/15326349.2013.840144pmid: N/A

A rate control problem is addressed for a queueing system in heavy traffic. The arrival process is a stationary heavy-tailed On-Off process and service is done at a constant rate (controlled). With an infinite horizon discounted cost function, the main result shows the existence of an optimal rate and specifies a bound on this optimal rate. As a part of the analysis, we solve an approximating control problem driven by fractional Brownian motion. We also derive an asymptotic maximal bound on the second moment of the centered On-Off process, which is a key ingredient of the proof and is of independent interest.

Li, Shuanming; Huang, Fengjing; Jin, Can

doi: 10.1080/15326349.2013.847610pmid: N/A

By using a generating function inverse relationship formula, we derive explicit expressions for the joint distributions of three pairs of ruin related quantities in the compound binomial risk model, including the time and the number of claims when the surplus first hits a certain level; the duration of negative surplus and the number of claims between the time of ruin; and the time of recovery, the time of ruin, and the number of claims by the time of ruin. The moments and covariance between the three pairs of random variables are also investigated.