Valuing equity-linked death benefits in jump diffusion models

The paper is motivated by the valuation problem of guaranteed minimum death benefits in various equity-linked products. At the time of death, a benefit payment is due. It may depend not only on the price of a stock or stock fund at that time, but also on prior prices. The problem is to calculate the expected discounted value of the benefit payment. Because the distribution of the time of death can be approximated by a combination of exponential distributions, it suffices to solve the problem for an exponentially distributed time of death. The stock price process is assumed to be the exponential of a Brownian motion plus an independent compound Poisson process whose upward and downward jumps are modeled by combinations (or mixtures) of exponential distributions. Results for exponential stopping of a Lévy process are used to derive a series of closed-form formulas for call, put, lookback, and barrier options, dynamic fund protection, and dynamic withdrawal benefit with guarantee. We also discuss how barrier options can be used to model lapses and surrenders.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

On evaluating the accuracy and biological plausibility of diffusion MRI tractograms

In diffusion MRI, traditional tractography algorithms do not recover truly quantitative tractograms and the structural connectivity has to be estimated indirectly by counting the number of fiber tracts or averaging scalar maps along them. Recently, global and efficient methods have emerged to estimate more quantitative tractograms by combining tractography with local models for the diffusion signal, like the Convex Optimization Modeling for Microstructure Informed Tractography (COMMIT) framework. In this abstract, we show the importance of using both (i) proper multi-compartment diffusion models and (ii) adequate multi-shell acquisitions, in order to evaluate the accuracy and the biological plausibility of the tractograms.

Towards robust network based complex systems : from evolutionary cellular automata to biological models

Abstract Sitting between your past and your future doesn't mean you are in the present. Dakota Skye Complex systems science is an interdisciplinary field grouping under the same umbrella dynamical phenomena from social, natural or mathematical sciences. The emergence of a higher order organization or behavior, transcending that expected of the linear addition of the parts, is a key factor shared by all these systems. Most complex systems can be modeled as networks that represent the interactions amongst the system's components. In addition to the actual nature of the part's interactions, the intrinsic topological structure of underlying network is believed to play a crucial role in the remarkable emergent behaviors exhibited by the systems. Moreover, the topology is also a key a factor to explain the extraordinary flexibility and resilience to perturbations when applied to transmission and diffusion phenomena. In this work, we study the effect of different network structures on the performance and on the fault tolerance of systems in two different contexts. In the first part, we study cellular automata, which are a simple paradigm for distributed computation. Cellular automata are made of basic Boolean computational units, the cells; relying on simple rules and information from- the surrounding cells to perform a global task. The limited visibility of the cells can be modeled as a network, where interactions amongst cells are governed by an underlying structure, usually a regular one. In order to increase the performance of cellular automata, we chose to change its topology. We applied computational principles inspired by Darwinian evolution, called evolutionary algorithms, to alter the system's topological structure starting from either a regular or a random one. The outcome is remarkable, as the resulting topologies find themselves sharing properties of both regular and random network, and display similitudes Watts-Strogtz's small-world network found in social systems. Moreover, the performance and tolerance to probabilistic faults of our small-world like cellular automata surpasses that of regular ones. In the second part, we use the context of biological genetic regulatory networks and, in particular, Kauffman's random Boolean networks model. In some ways, this model is close to cellular automata, although is not expected to perform any task. Instead, it simulates the time-evolution of genetic regulation within living organisms under strict conditions. The original model, though very attractive by it's simplicity, suffered from important shortcomings unveiled by the recent advances in genetics and biology. We propose to use these new discoveries to improve the original model. Firstly, we have used artificial topologies believed to be closer to that of gene regulatory networks. We have also studied actual biological organisms, and used parts of their genetic regulatory networks in our models. Secondly, we have addressed the improbable full synchronicity of the event taking place on. Boolean networks and proposed a more biologically plausible cascading scheme. Finally, we tackled the actual Boolean functions of the model, i.e. the specifics of how genes activate according to the activity of upstream genes, and presented a new update function that takes into account the actual promoting and repressing effects of one gene on another. Our improved models demonstrate the expected, biologically sound, behavior of previous GRN model, yet with superior resistance to perturbations. We believe they are one step closer to the biological reality.

Testing new identity models and processes in French-speaking adolescents and emerging adults students

Developing a sense of identity is a crucial psychosocial task for young people. The purpose of this study was to evaluate identity development in French-speaking adolescents and emerging adults (in France and Switzerland) using a process-oriented model of identity formation including five dimensions (i.e., exploration in breadth, commitment making, exploration in depth, identification with commitment, and ruminative exploration). The study included participants from three different samples (total N = 2239, 66.7% women): two samples of emerging adult student and one sample of adolescents. Results confirmed the hypothesized five-factor dimensional model of identity in our three samples and provided evidence for convergent validity of the model. The results also indicated that exploration in depth might be subdivided in two aspects: a first form of exploration in depth leading to a better understanding and to an increase of the strength of current commitments and a second form of exploration in depth leading to a re-evaluation and a reconsideration of current commitments. Further, the identity status cluster solution that emerged is globally in line with previous literature (i.e., achievement, foreclosure, moratorium, carefree diffusion, diffused diffusion, undifferentiated). However, despite a structural similarity, we found variations in identity profiles because identity development is shaped by cultural context. These specific variations are discussed in light of social, educational and economic differences between France and the French-speaking part of Switzerland. Implications and suggestions for future research are offered.

A receptor-like kinase mutant with absent endodermal diffusion barrier displays selective nutrient homeostasis defects.

The endodermis represents the main barrier to extracellular diffusion in plant roots, and it is central to current models of plant nutrient uptake. Despite this, little is known about the genes setting up this endodermal barrier. In this study, we report the identification and characterization of a strong barrier mutant, schengen3 (sgn3). We observe a surprising ability of the mutant to maintain nutrient homeostasis, but demonstrate a major defect in maintaining sufficient levels of the macronutrient potassium. We show that SGN3/GASSHO1 is a receptor-like kinase that is necessary for localizing CASPARIAN STRIP DOMAIN PROTEINS (CASPs)--major players of endodermal differentiation--into an uninterrupted, ring-like domain. SGN3 appears to localize into a broader band, embedding growing CASP microdomains. The discovery of SGN3 strongly advances our ability to interrogate mechanisms of plant nutrient homeostasis and provides a novel actor for localized microdomain formation at the endodermal plasma membrane.

The effects of physiologically plausible connectivity structure on local and global dynamics in large scale brain models.

Functionally relevant large scale brain dynamics operates within the framework imposed by anatomical connectivity and time delays due to finite transmission speeds. To gain insight on the reliability and comparability of large scale brain network simulations, we investigate the effects of variations in the anatomical connectivity. Two different sets of detailed global connectivity structures are explored, the first extracted from the CoCoMac database and rescaled to the spatial extent of the human brain, the second derived from white-matter tractography applied to diffusion spectrum imaging (DSI) for a human subject. We use the combination of graph theoretical measures of the connection matrices and numerical simulations to explicate the importance of both connectivity strength and delays in shaping dynamic behaviour. Our results demonstrate that the brain dynamics derived from the CoCoMac database are more complex and biologically more realistic than the one based on the DSI database. We propose that the reason for this difference is the absence of directed weights in the DSI connectivity matrix.

Interpretation of primary care physicians' attitude regarding rotavirus immunisation using diffusion of innovation theories.

To evaluate primary care physicians' attitude towards implementation of rotavirus (RV) immunisation into the Swiss immunisation schedule, an eight-question internet-based questionnaire was sent to the 3799 subscribers of InfoVac, a nationwide web-based expert network on immunisation issues, which reaches >95% of paediatricians and smaller proportions of other primary care physicians. Five demographic variables were also inquired. Descriptive statistics and multivariate analyses for the main outcome "acceptance of routine RV immunisation" and other variables were performed. Diffusion of innovation theory was used for data assessment. Nine-hundred seventy-seven questionnaires were returned (26%). Fifty percent of participants were paediatricians. Routine RV immunisation was supported by 146 participants (15%; so called early adopters), dismissed by 620 (64%), leaving 211 (21%) undecided. However, when asked whether they would recommend RV vaccination to parents if it were officially recommended by the federal authorities and reimbursed, 467 (48.5%; so called early majority) agreed to recommend RV immunisation. Multivariate analysis revealed that physicians who would immunise their own child (OR: 5.1; 95% CI: 4.1-6.3), hospital-based physicians (OR: 1.6; 95% CI: 1.1-2.3) and physicians from the French (OR: 1.6; 95% CI: 1.2-2.3) and Italian speaking areas of Switzerland (OR: 2.5; 95% CI: 1.1-5.8) were more likely to support RV immunisation. Diffusion of innovation theory predicts a >80% implementation if approximately 50% of a given population support an innovation. Introduction of RV immunisation in Switzerland is likely to be successful, if (i) the federal authorities issue an official recommendation and (ii) costs are covered by basic health care insurance.

Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data.

Microstructure imaging from diffusion magnetic resonance (MR) data represents an invaluable tool to study non-invasively the morphology of tissues and to provide a biological insight into their microstructural organization. In recent years, a variety of biophysical models have been proposed to associate particular patterns observed in the measured signal with specific microstructural properties of the neuronal tissue, such as axon diameter and fiber density. Despite very appealing results showing that the estimated microstructure indices agree very well with histological examinations, existing techniques require computationally very expensive non-linear procedures to fit the models to the data which, in practice, demand the use of powerful computer clusters for large-scale applications. In this work, we present a general framework for Accelerated Microstructure Imaging via Convex Optimization (AMICO) and show how to re-formulate this class of techniques as convenient linear systems which, then, can be efficiently solved using very fast algorithms. We demonstrate this linearization of the fitting problem for two specific models, i.e. ActiveAx and NODDI, providing a very attractive alternative for parameter estimation in those techniques; however, the AMICO framework is general and flexible enough to work also for the wider space of microstructure imaging methods. Results demonstrate that AMICO represents an effective means to accelerate the fit of existing techniques drastically (up to four orders of magnitude faster) while preserving accuracy and precision in the estimated model parameters (correlation above 0.9). We believe that the availability of such ultrafast algorithms will help to accelerate the spread of microstructure imaging to larger cohorts of patients and to study a wider spectrum of neurological disorders.

Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond.

The complex structural organization of the white matter of the brain can be depicted in vivo in great detail with advanced diffusion magnetic resonance (MR) imaging schemes. Diffusion MR imaging techniques are increasingly varied, from the simplest and most commonly used technique-the mapping of apparent diffusion coefficient values-to the more complex, such as diffusion tensor imaging, q-ball imaging, diffusion spectrum imaging, and tractography. The type of structural information obtained differs according to the technique used. To fully understand how diffusion MR imaging works, it is helpful to be familiar with the physical principles of water diffusion in the brain and the conceptual basis of each imaging technique. Knowledge of the technique-specific requirements with regard to hardware and acquisition time, as well as the advantages, limitations, and potential interpretation pitfalls of each technique, is especially useful.

Mechanisms underlying functional recovery after in vivo focal cerebral ischemia : from preconditioning to diffusion MRI

Version abregée L'ischémie cérébrale est la troisième cause de mort dans les pays développés, et la maladie responsable des plus sérieux handicaps neurologiques. La compréhension des bases moléculaires et anatomiques de la récupération fonctionnelle après l'ischémie cérébrale est donc extrêmement importante et représente un domaine d'intérêt crucial pour la recherche fondamentale et clinique. Durant les deux dernières décennies, les chercheurs ont tenté de combattre les effets nocifs de l'ischémie cérébrale à l'aide de substances exogènes qui, bien que testées avec succès dans le domaine expérimental, ont montré un effet contradictoire dans l'application clinique. Une approche différente mais complémentaire est de stimuler des mécanismes intrinsèques de neuroprotection en utilisant le «modèle de préconditionnement» : une brève insulte protège contre des épisodes d'ischémie plus sévères à travers la stimulation de voies de signalisation endogènes qui augmentent la résistance à l'ischémie. Cette approche peut offrir des éléments importants pour clarifier les mécanismes endogènes de neuroprotection et fournir de nouvelles stratégies pour rendre les neurones et la glie plus résistants à l'attaque ischémique cérébrale. Dans un premier temps, nous avons donc étudié les mécanismes de neuroprotection intrinsèques stimulés par la thrombine, un neuroprotecteur «préconditionnant» dont on a montré, à l'aide de modèles expérimentaux in vitro et in vivo, qu'il réduit la mort neuronale. En appliquant une technique de microchirurgie pour induire une ischémie cérébrale transitoire chez la souris, nous avons montré que la thrombine peut stimuler les voies de signalisation intracellulaire médiées par MAPK et JNK par une approche moléculaire et l'analyse in vivo d'un inhibiteur spécifique de JNK (L JNK) .Nous avons également étudié l'impact de la thrombine sur la récupération fonctionnelle après une attaque et avons pu démontrer que ces mécanismes moléculaires peuvent améliorer la récupération motrice. La deuxième partie de cette étude des mécanismes de récupération après ischémie cérébrale est basée sur l'investigation des bases anatomiques de la plasticité des connections cérébrales, soit dans le modèle animal d'ischémie transitoire, soit chez l'homme. Selon des résultats précédemment publiés par divers groupes ,nous savons que des mécanismes de plasticité aboutissant à des degrés divers de récupération fonctionnelle sont mis enjeu après une lésion ischémique. Le résultat de cette réorganisation est une nouvelle architecture fonctionnelle et structurelle, qui varie individuellement selon l'anatomie de la lésion, l'âge du sujet et la chronicité de la lésion. Le succès de toute intervention thérapeutique dépendra donc de son interaction avec la nouvelle architecture anatomique. Pour cette raison, nous avons appliqué deux techniques de diffusion en résonance magnétique qui permettent de détecter les changements de microstructure cérébrale et de connexions anatomiques suite à une attaque : IRM par tenseur de diffusion (DT-IR1V) et IRM par spectre de diffusion (DSIRM). Grâce à la DT-IRM hautement sophistiquée, nous avons pu effectuer une étude de follow-up à long terme chez des souris ayant subi une ischémie cérébrale transitoire, qui a mis en évidence que les changements microstructurels dans l'infarctus ainsi que la modification des voies anatomiques sont corrélés à la récupération fonctionnelle. De plus, nous avons observé une réorganisation axonale dans des aires où l'on détecte une augmentation d'expression d'une protéine de plasticité exprimée dans le cône de croissance des axones (GAP-43). En appliquant la même technique, nous avons également effectué deux études, rétrospective et prospective, qui ont montré comment des paramètres obtenus avec DT-IRM peuvent monitorer la rapidité de récupération et mettre en évidence un changement structurel dans les voies impliquées dans les manifestations cliniques. Dans la dernière partie de ce travail, nous avons décrit la manière dont la DS-IRM peut être appliquée dans le domaine expérimental et clinique pour étudier la plasticité cérébrale après ischémie. Abstract Ischemic stroke is the third leading cause of death in developed countries and the disease responsible for the most serious long-term neurological disability. Understanding molecular and anatomical basis of stroke recovery is, therefore, extremely important and represents a major field of interest for basic and clinical research. Over the past 2 decades, much attention has focused on counteracting noxious effect of the ischemic insult with exogenous substances (oxygen radical scavengers, AMPA and NMDA receptor antagonists, MMP inhibitors etc) which were successfully tested in the experimental field -but which turned out to have controversial effects in clinical trials. A different but complementary approach to address ischemia pathophysiology and treatment options is to stimulate and investigate intrinsic mechanisms of neuroprotection using the "preconditioning effect": applying a brief insult protects against subsequent prolonged and detrimental ischemic episodes, by up-regulating powerful endogenous pathways that increase resistance to injury. We believe that this approach might offer an important insight into the molecular mechanisms responsible for endogenous neuroprotection. In addition, results from preconditioning model experiment may provide new strategies for making brain cells "naturally" more resistant to ischemic injury and accelerate their rate of functional recovery. In the first part of this work, we investigated down-stream mechanisms of neuroprotection induced by thrombin, a well known neuroprotectant which has been demonstrated to reduce stroke-induced cell death in vitro and in vivo experimental models. Using microsurgery to induce transient brain ischemia in mice, we showed that thrombin can stimulate both MAPK and JNK intracellular pathways through a molecular biology approach and an in vivo analysis of a specific kinase inhibitor (L JNK1). We also studied thrombin's impact on functional recovery demonstrating that these molecular mechanisms could enhance post-stroke motor outcome. The second part of this study is based on investigating the anatomical basis underlying connectivity remodeling, leading to functional improvement after stroke. To do this, we used both a mouse model of experimental ischemia and human subjects with stroke. It is known from previous data published in literature, that the brain adapts to damage in a way that attempts to preserve motor function. The result of this reorganization is a new functional and structural architecture, which will vary from patient to patient depending on the anatomy of the damage, the biological age of the patient and the chronicity of the lesion. The success of any given therapeutic intervention will depend on how well it interacts with this new architecture. For this reason, we applied diffusion magnetic resonance techniques able to detect micro-structural and connectivity changes following an ischemic lesion: diffusion tensor MRI (DT-MRI) and diffusion spectrum MRI (DS-MRI). Using DT-MRI, we performed along-term follow up study of stroke mice which showed how diffusion changes in the stroke region and fiber tract remodeling is correlating with stroke recovery. In addition, axonal reorganization is shown in areas of increased plasticity related protein expression (GAP 43, growth axonal cone related protein). Applying the same technique, we then performed a retrospective and a prospective study in humans demonstrating how specific DTI parameters could help to monitor the speed of recovery and show longitudinal changes in damaged tracts involved in clinical symptoms. Finally, in the last part of this study we showed how DS-MRI could be applied both to experimental and human stroke and which perspectives it can open to further investigate post stroke plasticity.

Simulation of stromatolite growth using diffusion-limited aggregation and a discrete model of biogeochemical exchanges in microbial mat

Resume : Mieux comprendre les stromatolithes et les tapis microbiens est un sujet important en biogéosciences puisque cela aide à l'étude des premières formes de vie sur Terre, a mieux cerner l'écologie des communautés microbiennes et la contribution des microorganismes a la biominéralisation, et même à poser certains fondements dans les recherches en exobiologie. D'autre part, la modélisation est un outil puissant utilisé dans les sciences naturelles pour appréhender différents phénomènes de façon théorique. Les modèles sont généralement construits sur un système d'équations différentielles et les résultats sont obtenus en résolvant ce système. Les logiciels disponibles pour implémenter les modèles incluent les logiciels mathématiques et les logiciels généraux de simulation. L'objectif principal de cette thèse est de développer des modèles et des logiciels pour aider a comprendre, via la simulation, le fonctionnement des stromatolithes et des tapis microbiens. Ces logiciels ont été développés en C++ en ne partant d'aucun pré-requis de façon a privilégier performance et flexibilité maximales. Cette démarche permet de construire des modèles bien plus spécifiques et plus appropriés aux phénomènes a modéliser. Premièrement, nous avons étudié la croissance et la morphologie des stromatolithes. Nous avons construit un modèle tridimensionnel fondé sur l'agrégation par diffusion limitée. Le modèle a été implémenté en deux applications C++: un moteur de simulation capable d'exécuter un batch de simulations et de produire des fichiers de résultats, et un outil de visualisation qui permet d'analyser les résultats en trois dimensions. Après avoir vérifié que ce modèle peut en effet reproduire la croissance et la morphologie de plusieurs types de stromatolithes, nous avons introduit un processus de sédimentation comme facteur externe. Ceci nous a mené a des résultats intéressants, et permis de soutenir l'hypothèse que la morphologie des stromatolithes pourrait être le résultat de facteurs externes autant que de facteurs internes. Ceci est important car la classification des stromatolithes est généralement fondée sur leur morphologie, imposant que la forme d'un stromatolithe est dépendante de facteurs internes uniquement (c'est-à-dire les tapis microbiens). Les résultats avancés dans ce mémoire contredisent donc ces assertions communément admises. Ensuite, nous avons décidé de mener des recherches plus en profondeur sur les aspects fonctionnels des tapis microbiens. Nous avons construit un modèle bidimensionnel de réaction-diffusion fondé sur la simulation discrète. Ce modèle a été implémenté dans une application C++ qui permet de paramétrer et exécuter des simulations. Nous avons ensuite pu comparer les résultats de simulation avec des données du monde réel et vérifier que le modèle peut en effet imiter le comportement de certains tapis microbiens. Ainsi, nous avons pu émettre et vérifier des hypothèses sur le fonctionnement de certains tapis microbiens pour nous aider à mieux en comprendre certains aspects, comme la dynamique des éléments, en particulier le soufre et l'oxygène. En conclusion, ce travail a abouti à l'écriture de logiciels dédiés à la simulation de tapis microbiens d'un point de vue tant morphologique que fonctionnel, suivant deux approches différentes, l'une holistique, l'autre plus analytique. Ces logiciels sont gratuits et diffusés sous licence GPL (General Public License). Abstract : Better understanding of stromatolites and microbial mats is an important topic in biogeosciences as it helps studying the early forms of life on Earth, provides clues re- garding the ecology of microbial ecosystems and their contribution to biomineralization, and gives basis to a new science, exobiology. On the other hand, modelling is a powerful tool used in natural sciences for the theoretical approach of various phenomena. Models are usually built on a system of differential equations and results are obtained by solving that system. Available software to implement models includes mathematical solvers and general simulation software. The main objective of this thesis is to develop models and software able to help to understand the functioning of stromatolites and microbial mats. Software was developed in C++ from scratch for maximum performance and flexibility. This allows to build models much more specific to a phenomenon rather than general software. First, we studied stromatolite growth and morphology. We built a three-dimensional model based on diffusion-limited aggregation. The model was implemented in two C++ applications: a simulator engine, which can run a batch of simulations and produce result files, and a Visualization tool, which allows results to be analysed in three dimensions. After verifying that our model can indeed reproduce the growth and morphology of several types of stromatolites, we introduced a sedimentation process as an external factor. This lead to interesting results, and allowed to emit the hypothesis that stromatolite morphology may be the result of external factors as much as internal factors. This is important as stromatolite classification is usually based on their morphology, imposing that a stromatolite shape is dependant on internal factors only (i.e. the microbial mat). This statement is contradicted by our findings, Second, we decided to investigate deeper the functioning of microbial mats, We built a two-dimensional reaction-diffusion model based on discrete simulation, The model was implemented in a C++ application that allows setting and running simulations. We could then compare simulation results with real world data and verify that our model can indeed mimic the behaviour of some microbial mats. Thus, we have proposed and verified hypotheses regarding microbial mats functioning in order to help to better understand them, e.g. the cycle of some elements such as oxygen or sulfur. ln conclusion, this PhD provides a simulation software, dealing with two different approaches. This software is free and available under a GPL licence.

Analysis of a finite volume method for a cross-diffusion model in population dynamics

The main goal of this paper is to propose a convergent finite volume method for a reactionâeuro"diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.

Intravoxel incoherent motion diffusion-weighted imaging in the liver: comparison of mono-, bi- and tri-exponential modelling at 3.0-T.

PURPOSE: To determine whether a mono-, bi- or tri-exponential model best fits the intravoxel incoherent motion (IVIM) diffusion-weighted imaging (DWI) signal of normal livers. MATERIALS AND METHODS: The pilot and validation studies were conducted in 38 and 36 patients with normal livers, respectively. The DWI sequence was performed using single-shot echoplanar imaging with 11 (pilot study) and 16 (validation study) b values. In each study, data from all patients were used to model the IVIM signal of normal liver. Diffusion coefficients (Di ± standard deviations) and their fractions (fi ± standard deviations) were determined from each model. The models were compared using the extra sum-of-squares test and information criteria. RESULTS: The tri-exponential model provided a better fit than both the bi- and mono-exponential models. The tri-exponential IVIM model determined three diffusion compartments: a slow (D1 = 1.35 ± 0.03 × 10(-3) mm(2)/s; f1 = 72.7 ± 0.9 %), a fast (D2 = 26.50 ± 2.49 × 10(-3) mm(2)/s; f2 = 13.7 ± 0.6 %) and a very fast (D3 = 404.00 ± 43.7 × 10(-3) mm(2)/s; f3 = 13.5 ± 0.8 %) diffusion compartment [results from the validation study]. The very fast compartment contributed to the IVIM signal only for b values ≤15 s/mm(2) CONCLUSION: The tri-exponential model provided the best fit for IVIM signal decay in the liver over the 0-800 s/mm(2) range. In IVIM analysis of normal liver, a third very fast (pseudo)diffusion component might be relevant. KEY POINTS: ? For normal liver, tri-exponential IVIM model might be superior to bi-exponential ? A very fast compartment (D = 404.00 ± 43.7 × 10 (-3)  mm (2) /s; f = 13.5 ± 0.8 %) is determined from the tri-exponential model ? The compartment contributes to the IVIM signal only for b ≤ 15 s/mm (2.)