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.

Using Normative Markov Decision Processes for Evaluating Electronic Contracts: A Case Study in a Simulated Aerospace Aftermarket

Before signing electronic contracts, a rational agent should estimate the expected utilities of these contracts and calculate the violation risks related to them. In order to perform such pre-signing procedures, this agent has to be capable of computing a policy taking into account the norms and sanctions in the contracts. In relation to this, the contribution of this work is threefold. First, we present the Normative Markov Decision Process, an extension of the Markov Decision Process for explicitly representing norms. In order to illustrate the usage of our framework, we model an example in a simulated aerospace aftermarket. Second, we specify an algorithm for identifying the states of the process which characterize the violation of norms. Finally, we show how to compute policies with our framework and how to calculate the risk of violating the norms in the contracts by adopting a particular policy.

Average control of Markov decision processes with Feller transition probabilities and general action spaces

This paper studies the average control problem of discrete-time Markov Decision Processes (MDPs for short) with general state space, Feller transition probabilities, and possibly non-compact control constraint sets A(x). Two hypotheses are considered: either the cost function c is strictly unbounded or the multifunctions A(r)(x) = {a is an element of A(x) : c(x, a) <= r} are upper-semicontinuous and compact-valued for each real r. For these two cases we provide new results for the existence of a solution to the average-cost optimality equality and inequality using the vanishing discount approach. We also study the convergence of the policy iteration approach under these conditions. It should be pointed out that we do not make any assumptions regarding the convergence and the continuity of the limit function generated by the sequence of relative difference of the alpha-discounted value functions and the Poisson equations as often encountered in the literature. (C) 2012 Elsevier Inc. All rights reserved.

SINGULARLY PERTURBED DISCOUNTED MARKOV CONTROL PROCESSES IN A GENERAL STATE SPACE

This paper studies the asymptotic optimality of discrete-time Markov decision processes (MDPs) with general state space and action space and having weak and strong interactions. By using a similar approach as developed by Liu, Zhang, and Yin [Appl. Math. Optim., 44 (2001), pp. 105-129], the idea in this paper is to consider an MDP with general state and action spaces and to reduce the dimension of the state space by considering an averaged model. This formulation is often described by introducing a small parameter epsilon > 0 in the definition of the transition kernel, leading to a singularly perturbed Markov model with two time scales. Our objective is twofold. First it is shown that the value function of the control problem for the perturbed system converges to the value function of a limit averaged control problem as epsilon goes to zero. In the second part of the paper, it is proved that a feedback control policy for the original control problem defined by using an optimal feedback policy for the limit problem is asymptotically optimal. Our work extends existing results of the literature in the following two directions: the underlying MDP is defined on general state and action spaces and we do not impose strong conditions on the recurrence structure of the MDP such as Doeblin's condition.

Cooperative off-policy prediction of markov decision processes in adaptive networks

We apply diffusion strategies to propose a cooperative reinforcement learning algorithm, in which agents in a network communicate with their neighbors to improve predictions about their environment. The algorithm is suitable to learn off-policy even in large state spaces. We provide a mean-square-error performance analysis under constant step-sizes. The gain of cooperation in the form of more stability and less bias and variance in the prediction error, is illustrated in the context of a classical model. We show that the improvement in performance is especially significant when the behavior policy of the agents is different from the target policy under evaluation.

Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.

On parameter estimation in population models

We describe methods for estimating the parameters of Markovian population processes in continuous time, thus increasing their utility in modelling real biological systems. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to a class of models called density-dependent Markov population processes. We illustrate the versatility of both approaches by estimating the parameters of the stochastic SIS logistic model from simulated data. This model is also fitted to data from a population of Bay checkerspot butterfly (Euphydryas editha bayensis), allowing us to assess the viability of this population. (c) 2006 Elsevier Inc. All rights reserved.

Lp Microlocal Supercritical Markov Branching Processes with Non-Homogeneous Poisson Immigration

2010 Mathematics Subject Classification: 60J80.

Likelihood-Based Inference for Partially Observed Multi-Type Markov Branching Processes

Thesis (Ph.D.)--University of Washington, 2016-08

A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.

Stochastic models for mainland-island metapopulations in static and dynamic landscapes

This paper has three primary aims: to establish an effective means for modelling mainland-island metapopulations inhabiting a dynamic landscape: to investigate the effect of immigration and dynamic changes in habitat on metapopulation patch occupancy dynamics; and to illustrate the implications of our results for decision-making and population management. We first extend the mainland-island metapopulation model of Alonso and McKane [Bull. Math. Biol. 64:913-958,2002] to incorporate a dynamic landscape. It is shown, for both the static and the dynamic landscape models, that a suitably scaled version of the process converges to a unique deterministic model as the size of the system becomes large. We also establish that. under quite general conditions, the density of occupied patches, and the densities of suitable and occupied patches, for the respective models, have approximate normal distributions. Our results not only provide us with estimates for the means and variances that are valid at all stages in the evolution of the population, but also provide a tool for fitting the models to real metapopulations. We discuss the effect of immigration and habitat dynamics on metapopulations, showing that mainland-like patches heavily influence metapopulation persistence, and we argue for adopting measures to increase connectivity between this large patch and the other island-like patches. We illustrate our results with specific reference to examples of populations of butterfly and the grasshopper Bryodema tuberculata.

Path integrals for continuous-time Markov chains

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

Discrete time Non-homogeneous Semi-Markov Processes applied to Models for Disability Insurance

In this paper, we present a stochastic model for disability insurance contracts. The model is based on a discrete time non-homogeneous semi-Markov process (DTNHSMP) to which the backward recurrence time process is introduced. This permits a more exhaustive study of disability evolution and a more efficient approach to the duration problem. The use of semi-Markov reward processes facilitates the possibility of deriving equations of the prospective and retrospective mathematical reserves. The model is applied to a sample of contracts drawn at random from a mutual insurance company.

Low dispersing species : population genetic processes in space and time : the genus "Trochulus" (Gastropoda: Hygromiidae) as a case study

RÉSUMÉ Une espèce est rarement composée d'une population unique. Parce que les individus ont des capacités de dispersion limitées et que les paysages sont des mosaïques d'habitats, la plupart des espèces sont plutôt composées de sous-populations connectées par la migration. Cette variation spatiale influence directement la distribution de la variabilité génétique dans et entre les populations. Durant ce travail, nous avons abordé certains des processus populationnels qui ont joué un rôle supposé dans l'apparition de nouvelles espèces au sein du genre Trochulus. Plus précisément, nous avons tenté d'évaluer les impacts respectifs de l'isolement passé (facteurs historiques) et présent (facteurs locaux). Nous avons d'abord pu montrer que les faibles capacités de dispersion des escargots terrestres ont directement influencé leur histoire évolutive à toutes les échelles spatiales et temporelles. En réduisant l'effet homogénéisant de la migration, une faible dispersion maintient dans les populations les traces génétiques d'évènements passés. A l'échelle de la distribution globale de Trochulus villosus, ces traces ont permis de reconstruire une histoire faite d'isolements et d'expansions de populations. En combinant des données génétiques avec une modélisation de la niche climatique passée, il a été possible de proposer un scénario significativement meilleur que toutes les hypothèses alternatives que nous avons testées. A l'échelle locale par contre, l'héritage historique est difficile à distinguer de la dynamique actuelle. Ce fut le cas des lignées mitochondriales du complexe sericeus-hispidus : les deux principales lignées étaient phylogénétiquement éloignées, avaient eu des démographies passées différentes et corrélaient avec des différences morphologiques. D'un autre côté, le flux de gène nucléaire était fort, contredisant l'idée de deux espèces cryptiques isolées reproductivement. Pour pouvoir conclure à la présence ou non de deux espèces, il nous a manqué des informations locales sur la dynamique des populations et les conditions écologiques que l'on trouve dans la région d'étude. Enfin, nous avons pu souligner que la connectivité entre populations d'escargots est soumise à la qualité des habitats et à leur organisation spatiale. Les escargots sont dépendants d'un habitat et s'y adaptent, comme l'indiquent la présence de «poils » uniquement sur la coquille d'espèces vivant dans des habitats humides ou la corrélation entre morphologie et habitat au sein du complexe sericeus-hispidus. Logiquement donc, les escargots migrent préférentiellement au travers d'habitats favorables comme l'a montré la réduction de flux de gènes au travers des prairies chez T. villosus (une espèce forestière). De ces données, nous pouvons supposer que les populations d'escargots en particulier, et des espèces à faible dispersion en général, ont de fortes chances d'être affectées par les changements climatiques, avec de probables implications pour leurs histoires évolutives. SUMMARY : Species rarely consists in a single population. Because individuals have limited dispersal abilities, because landscapes are habitat patchworks, most species are made of several subpopulations connected by migration. This spatial variation has consequences on the distribution of genetic diversity within and between populations, creating a structure among the populations. During the present work, we investigated some of the population processes assumed to have played an important role on the speciation within the genus Trochulus. More specifically, we questioned the respective impacts of past (historical factors) or present (local factors) population isolations. We first could show that the poor dispersal abilities of land snails have had profound impacts on their evolutionary histories at all spatial and temporal scales. Low dispersal maintains a strong signature of past events in the populations by minimising the homogenising effects of geneflow. At the scale of Trochulus villosus global distribution, they allowed to retrieve the detailed history of this species population isolations and expansions. Combining a large genetic dataset with paleo-climatic niche modelling ended up with a historical scenario significantly better than all traditional alternatives we tested. At local scale on the contrary, past events become difficult to tease apart from ongoing processes. This was the case for the divergent mitochondria) lineages within the sericeus-hispidus complex: the two principal lineages appeared to be phylogenetically distant, to have experienced different demographic histories and to correlate with morphological differences. On the other hand, nuclear (present day) geneflow was high, contradicting the idea of two reproductively isolated cryptic species. Information on the local population dynamics and environmental conditions are lacking to be able to decide whether past isolation has indeed resulted here in new species. Finally, we emphasised the importance of the habitat types present in a landscape as well as their spatial organisation for the population connectivity of land snails. These species are tightly dependent on a habitat and adapt to it as shown by thé occurrence of hair-like structures only in species living in humid environments or by the correlation between shell morphology and habitat in the sericeus-hispidus complex. As a result, land snails preferentially migrate through favourable habitats: Trochulus villosus, a forest species, had its geneflow significantly reduced across meadows. From these data, we can hypothesise that the populations of land snails in particular and of low dispersing species in general are likely to be strongly affected by the ongoing climate changes, with potential major consequences on their evolutionary histories.