Pricing zero-coupon CAT bonds using the enlargement of ltration theory: a general framework

The main goal of this paper is to use the enlargement of ltration framework for pricing zerocoupon CAT bonds. For this purpose, we develop two models where the trigger event time is perfectly covered by an increasing sequence of stopping times with respect to a reference ltration. Hence, depending on the nature of these stopping times the trigger event time can be either accessible or totally inaccessible. When some of these stopping times are not predictable, the trigger event time is totally inaccessible, and very nice mathematical computations can be derived. When the stopping times are predictable, the trigger event time is accessible, and this case would be a meaningful choice for Model 1 from a practical point of view since features like seasonality are already captured by some quantities such as the stochastic intensity of the Poisson process. We compute the main tools for pricing the zero-coupon CAT bond and show that our constructions are more general than some existing models in the literature. We obtain some closed-form prices of zero-coupon CAT bonds in Model 2 so we give a numerical illustrative example for this latter.Comment: Journal of Mathematical Finance, Scientific Research, In pres

Queueing models for capacity changes in cellular networks

With the rapid development of cellular communication techniques, many recent studies have focused on improving the quality of service (QoS) in cellular networks. One characteristic of the systems in cellular networks, which can have direct impact on the system QoS, is the fluctuation of the system capacity. In this thesis, the QoS of systems with capacity fluctuations is studied from two perspectives: (1) priority queueing systems with preemption, and (2) the M/M/~C/~C system. In the first part, we propose two models with controlled preemption and analyze their performance in the context of a single reference cell that supports two kinds of traffic (new calls and handoff calls). The formulae for calculating the performance measures of interest (i.e., handoff call blocking probability, new call blocking and dropping probabilities) are developed, and the procedures for solving optimization problems for the optimal number of channels required for each proposed model are established. The proposed controlled preemption models are then compared to existing non-preemption and full preemption models from the following three perspectives: (i) channel utilization, (ii) low priority call (i.e., new calls) performance, and (iii) flexibility to meet various constraints. The results showed that the proposed controlled preemption models are the best models overall. In the second part, the loss system with stochastic capacity, denoted by M/M/~C/~C, is analyzed using the Markov regenerative process (MRGP) method. Three different distributions of capacity interchange times (exponential, gamma, and Pareto) and three different capacity variation patterns (skip-free, distance-based, and uniform-based) are considered. Analytic expressions are derived to calculate call blocking and dropping probabilities and are verified by call level simulations. Finally, numerical examples are provided to determine the impact of different distributions of capacity interchange times and different capacity variation patterns on system performance

Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)

Analyse mathématique, méthode de calcul de la gigue et applications aux réseaux Internet

RÉSUMÉ Internet, ces dernières années, sert de support de communication à un grand nombre d’applications. L’évolution des réseaux à haut débit ont facilité le progrès des applications multimédia comme la voix sur IP, la vidéo streaming ou la vidéo interactive en temps réel... La variation de la disponibilité des ressources du réseau ne peut pas garantir une bonne qualité à tout moment pour ces services. C’est dans ce contexte que les travaux de ce projet de doctorat s’inscrivent et précisément dans le cadre de l’optimisation de la qualité de service (QoS). Les mécanismes de contrôle de QoS sont variés. On retrouve le contrôle de délai, assuré par la stratégie d’ordonnancement des paquets. Le contrôle de débit, quant à lui, fait en sorte que le débit de la source soit égal à la bande passante disponible dans le réseau. Excepté que les applications vidéo, surtout en temps réel, sont très sensibles à la variation du délai, appelée la gigue. En effet, la qualité perçue par les clients des vidéos en ligne dépend étroitement de la gigue. Une augmentation de la gigue engendre principalement des problèmes de démarrage retardé de la vidéo, des interruptions au cours de la vidéo et des distorsions de la résolution. L’objectif de cette thèse est d’étudier le paramètre de la gigue, qui demeure peu étudiée dans la littérature sur les réseaux IP, ainsi que d’envisager l’impact de l’augmentation de ce paramètre sur la vidéo transmise sur IP, l’une des applications les plus populaires de nos jours. Toutefois, au-delà des difficultés de la modélisation du trafic et du réseau, cet objectif majeur pose de nombreuses problématiques. Comment calculer la gigue analytiquement pour un trafic modélisé par des distributions généralisées au niveau paquet ? Est-ce que les modèles proposés sont suffisamment simples et faciles à calculer ? Comment intégrer ces nouvelles formalisations pour le contrôle des performances ? Comment l’estimation analytique peut- elle minimiser le trafic des paquets de contrôle des connexions vidéo? Nous explorons tout d’abord le calcul de la gigue dans des files d’attente avec des trafics autres que le trafic Poisson. Ce dernier est largement utilisé pour modéliser le trafic sur Internet étant donnée sa simplicité en échange de la imprécision. L’idée pour le calcul de la gigue est d’utiliser, d’une part la même formule que le cas du Poisson mais en intégrant d’autres distributions, et d’autre part des approximations et des hypothèses quand la caractérisation analytique du temps de transit n’est pas possible. Nous adoptons la simulation pour valider les modèles approximatifs. L’ensemble de simulations montre que la gigue moyenne calculée par notre modèle et celle obtenue par simulation coïncident avec des intervalles de confiance adéquats. De plus, le temps de calcul estimé pour évaluer la gigue est minime, ce qui facilite l’utilisation des formules proposées dans des outils de contrôle et en optimisation.-----------ABSTRACT In recent years, we have witnessed the huge use of the Internet Protocol for delivering multimedia trafic. Developments in broadband networks led the progress in multimedia applications such as voice over IP, video streaming or real-time videos. However, the stochastic nature of the networks, in particular mobile networks, make it difficult to maintain a good quality at all times. The research of this PhD thesis deals with the improvement of the quality of service (QoS) for this kind of applications. Current network protocols provide multiple QoS control mechanism. Congestion control and transmission delay optimization are provided by packet scheduling strategies and bandwidth planning. Moreover, flow control adjusts the mismatch between the video server rate and the receiver available bandwidth. Nevertheless, video applications, in particular interactive videos, are very sensitive to delay variation, commonly called jitter. Indeed, the customers’ perceived video quality depends on it. A jitter increase may cause a large video start-up delay, video interruptions and a decrease of image quality. The main objective of this thesis is the study of jitter, which is not much studied in the IP literature. We also examine the impact of the increase of this parameter on video transmission. However, beyond the difficulties of modeling traffic and network, this major objective raises many other issues. How to calculate jitter analytically for traffic models with general distributions? Are the proposed models sufficiently simple and easy to calculate? How to integrate these new formalizations into performance monitoring? How can the analytical estimate minimize the traffic control packets exchange for each video connection? We first explore the jitter calculation in queues with traffic other than Poisson traffic, that was widely used to model Internet traffic because of its simplicity. The idea is to compute jitter with the same formula for the Poisson traffic case, but with other distributions. For this, we need some approximations and assumptions when the analytical characterization of the transit time is not possible. We adopt simulations to validate the approximate models. The set of simulations shows that the average jitter calculated by our model and by simulation coincide within an appropriate confidence intervals. Moreover, the execution time to evaluate jitter is small, which facilitates the use of the proposed formulas in control tools and in optimization models. We then study the possibility of exploiting this analytical results to control jitter buffers, an important component in the video transmission. We find that it is possible to evaluate its performances analytically by estimating jitter inside this type of buffer