Multifractional Poisson process, multistable subordinator and related limit theorems

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and establish the convergence of a continuous-time random walk to the multifractional Poisson process.

Stochastic growth models for analyzing crustacean data

The contemporary methodology for growth models of organisms is based on continuous trajectories and thus it hinders us from modelling stepwise growth in crustacean populations. Growth models for fish are normally assumed to follow a continuous function, but a different type of model is needed for crustacean growth. Crustaceans must moult in order for them to grow. The growth of crustaceans is a discontinuous process due to the periodical shedding of the exoskeleton in moulting. The stepwise growth of crustaceans through the moulting process makes the growth estimation more complex. Stochastic approaches can be used to model discontinuous growth or what are commonly known as "jumps" (Figure 1). However, in stochastic growth model we need to ensure that the stochastic growth model results in only positive jumps. In view of this, we will introduce a subordinator that is a special case of a Levy process. A subordinator is a non-decreasing Levy process, that will assist in modelling crustacean growth for better understanding of the individual variability and stochasticity in moulting periods and increments. We develop the estimation methods for parameter estimation and illustrate them with the help of a dataset from laboratory experiments. The motivational dataset is from the ornate rock lobster, Panulirus ornatus, which can be found between Australia and Papua New Guinea. Due to the presence of sex effects on the growth (Munday et al., 2004), we estimate the growth parameters separately for each sex. Since all hard parts are shed too often, the exact age determination of a lobster can be challenging. However, the growth parameters for the aforementioned moult processes from tank data being able to estimate through: (i) inter-moult periods, and (ii) moult increment. We will attempt to derive a joint density, which is made up of two functions: one for moult increments and the other for time intervals between moults. We claim these functions are conditionally independent given pre-moult length and the inter-moult periods. The variables moult increments and inter-moult periods are said to be independent because of the Markov property or conditional probability. Hence, the parameters in each function can be estimated separately. Subsequently, we integrate both of the functions through a Monte Carlo method. We can therefore obtain a population mean for crustacean growth (e. g. red curve in Figure 1). [GRAPHICS]

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement, le quatrième chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Lévy model" écrit en collaboration avec Romuald Hervé Momeya et publié dans la revue Asia-Pacific Financial Market. Cet article présente de nouveaux résultats en lien avec le problème de l'incomplétude dans un marché financier où le processus de prix de l'actif risqué est décrit par un modèle exponentiel markovien additif. Ces résultats consistent à charactériser la mesure martingale satisfaisant le critère de l'entropie. Cette mesure est utilisée pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modèle exponentiel de Lévy avec changement de régimes.

Fractional Fokker-Planck-Kolmogorov type Equations and their Associated Stochastic Differential Equations

MSC 2010: 26A33, 35R11, 35R60, 35Q84, 60H10 Dedicated to 80-th anniversary of Professor Rudolf Gorenflo

Revisiting the origin and development of pleonastic that in English

The origin of pleonastic that can be traced back to Old English where it could appear in syntactic constructions consisting of a preposition + demonstrative pronoun (i.e. for þy þat, for þæm þe) or a subordinator (i.e. oþ þat). Its diffusion with other subordinators is considered an early Middle English development as a result of the standardization of this item as the general subordinator in the period, which motivated its use as a pleonastic word in combination with all kinds of conjunctions (i.e. now that, gif that, when that, etc.) and prepositions (i.e. before that, save that, in that). Its use considerably increased in late Middle English, declining throughout the 17th century. The list of subordinating elements includes relativizers (i.e. this that), adverbial relatives (i.e. there that) and a number of subordinators (i.e. after, as, because, before, beside, for, if, since, sith, though, until, when, while, etc.). The present paper pursues the following objectives: a) to analyse the use and distribution of pleonastic that in a corpus of early English medical writing (in the period 1375-1700); b) to classify the construction in terms of the two different varieties of medical texts, i.e. treatises and recipes; and c) to assess the decline of the construction with the different conjunctive words. The data used as sources of evidence come from The Corpus of Early English Medical Writing, i.e. Middle English Medical Texts (MEMT for the period 1375-1500) and Early Modern English Medical Texts (EMEMT for the period 1500-1700).