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Le travail développe un modèle de portfolio à propos de la dollarisation financière (FD), et l'estime pour la Russie. La contribution de ce travail sera de construire le premier modèle théorique de variance moyenne asymétrique d'aplatissement sur la dollarisation financière et de le valider empiriquement. Le travail se fonde sur des recherches antérieures qui ont trouvé que l'ajout de moments plus élevés, comme l'asymétrie et l'aplatissement, à la variance minimale du portfolio(MVP) permettant une meilleure modélisation des choix de portfolio et de développe un model comme celui-ci pour la FD. Nous utilisons ensuite les méthodes Markovswitching sur les données mensuelles pour les dépôts bancaires en Russie depuis la fin des années 1990 afin de documenter l'influence dominante de l'inflation et de la dépréciation de la monnaie et de leurs moments comme principaux déterminants de dépôt de dollarisation dans un cadre de variance-moyenne-asymétrique-aplatie en période de crise, par opposition aux périodes normales. / This thesis develops a portfolio model of financial dollarization (FD) and estimates it for Russia. The contribution of this work will be to construct the first theoretical meanvariance-skewness-kurtosis model of financial dollarization and to validate it empirically. The work builds on previous research which found that adding higher moments, as Skewness and Kurtosis, to the minimum variance portfolio (MVP) enables a better modelling of portfolio choice, and develops such a model for FD. We then use Markovswitching methods on monthly data for bank deposits in Russia since the late 1990s to document the dominant influence of inflation and currency depreciation and their moments as the main determinants of deposit dollarization in a mean-varianceskewness-kurtosis framework during crisis as opposed to normal periods.

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Thesis--University of Oklahoma. / Includes bibliographical references (leaf 57).

Markov chains offer a rigorous mathematical framework to describe systems that exhibit stochastic behaviour, as they are supported by a plethora of methodologies to analyse their properties. Stochastic process algebras are high-level formalisms, where systems are represented as collections of interacting components. This compositional approach to modelling allows us to describe complex Markov chains using a compact high-level specification. There is an increasing need to investigate the properties of complex systems, not only in the field of computer science, but also in computational biology. To explore the stochastic properties of large Markov chains is a demanding task in terms of computational resources. Approximating the stochastic properties can be an effective way to deal with the complexity of large models. In this thesis, we investigate methodologies to approximate the stochastic behaviour of Markovian process algebra models. The discussion revolves around two main topics: approximate state-space aggregation and stochastic simulation. Although these topics are different in nature, they are both motivated by the need to efficiently handle complex systems. Approximate Markov chain aggregation constitutes the formulation of a smaller Markov chain that approximates the behaviour of the original model. The principal hypothesis is that states that can be characterised as equivalent can be adequately represented as a single state. We discuss different notions of approximate state equivalence, and how each of these can be used as a criterion to partition the state-space accordingly. Nevertheless, approximate aggregation methods typically require an explicit representation of the transition matrix, a fact that renders them impractical for large models. We propose a compositional approach to aggregation, as a means to efficiently approximate complex Markov models that are defined in a process algebra specification, PEPA in particular. Regarding our contributions to Markov chain simulation, we propose an accelerated method that can be characterised as almost exact, in the sense that it can be arbitrarily precise. We discuss how it is possible to sample from the trajectory space rather than the transition space. This approach requires fewer random samples than a typical simulation algorithm. Most importantly, our approach does not rely on particular assumptions with respect to the model properties, in contrast to otherwise more efficient approaches.

La confiance dans les systèmes complexes est aujourd'hui primordiale. Parmi les langages de modélisation dysfonctionnelle des systèmes, les chaînes de Markov sont un bon compromis entre la calculabilité des modèles et le pouvoir d'expression qu'elles apportent. Cependant, comme les chaînes de Markov rendent compte des différents états du système, leur taille est confrontée à l'explosion combinatoire. Il y a deux obstacles majeurs induits par cette explosion : la difficulté d'écrire des chaînes pour les grands systèmes à la main, et les besoins en ressources calculatoires pour leur résolution. Le premier obstacle est dépassé facilement en compilant les chaînes de Markov depuis un modèle de plus haut niveau (dans ces travaux, AltaRica 3.0 est utilisé).Dans cette thèse, nous nous sommes concentrés sur la génération partielle de chaînes de Markov, afin de dépasser le problème d'explosion combinatoire. La méthode est fondée sur l'observation que les systèmes réparables, même les plus grands, passent leur temps dans un petit nombre d'états proches de l'état nominal du système. La génération partielle utilise l'algorithme de Dijkstra, auquel est combiné un facteur de pertinence, qui permet la sélection des états les plus probables du système. Il est possible d'encadrer les valeurs des indicateurs de sûreté obtenus avec la chaîne partielle grâce à l'introduction d'une chaîne partielle avec puits.La méthode de génération partielle est entièrement implémentée et fait partie du projet AltaRica 3.0. Il est ainsi possible de calculer les indicateurs de sûreté des systèmes directement depuis un modèle AltaRica. Divers expériences ont été menées pour illustrer la faisabilité de la méthode, son passage à l'échelle, ainsi que ses points forts et ses limites. / Trustworthiness in systems is of paramount importance. Among safety modeling languages, Markov chains are a good tradeoff between the safety concepts that can be modeled and the ease of calculation. However, as they model the different states of the systems, they suffer from the state space explosion. This explosion has two drawbacks: it makes Markov chains very difficult to write by hand for large systems, and large Markov chain calculation is resource consuming. The first drawback is easily tackled by generating Markov chains from higher-level languages (such as AltaRica 3.0).In this thesis, we focused on the partial generation of Markov chains, to tackle the state space explosion of the models. This idea is based on the observation that even large repairable systems spent most of their times in a few number of states, that are close to the nominal state of the system. The partial generation is based on Dijkstra's algorithm and on a so-called relevance factor to generate only the most probable states of the Markov chain. The reliability indicators obtained with such a partial chain can be bounded with a slightly different Markov chain.The partial generation method is fully implemented in the AltaRica 3.0 project to automatically calculate the reliability indicators of a system modeled in AltaRica. Different experiments illustrate the practability of the method, as well as its strengths and weaknesses.

The use of networks to model the spread of epidemics through structured populations is widespread. However, epidemics on networks lead to intractable exact systems with the need to coarse grain and focus on some average quantities. Often, the underlying stochastic processes are Markovian and so are the resulting mean-field models constructed as systems of ordinary differential equations (ODEs). However, the lack of memory (or memorylessness) does not accurately describe real disease dynamics. For instance, many epidemiological studies have shown that the true distribution of the infectious period is rather centred around its mean, whereas the memoryless assumption imposes an exponential distribution on the infectious period. Assumptions such as these greatly affect the predicted course of an epidemic and can lead to inaccurate predictions about disease spread. Such limitations of existing approaches to modelling epidemics on networks motivated my efforts to develop non-Markovian models which would be better suited to capture essential realistic features of disease dynamics. In the first part of my thesis I developed a pairwise, multi-stage SIR (susceptible-infected-recovered) model. Each infectious node goes through some K 2 N infectious stages, which for K > 1 means that the infectious period is gamma-distributed. Analysis of the model provided analytic expressions for the epidemic threshold and the expected final epidemic size. Using available epidemiological data on the infectious periods of various diseases, I demonstrated the importance of considering the shape of the infectious period distribution. The second part of the thesis expanded the framework of non-Markovian dynamics to networks with heterogeneous degree distributions with non-negligible levels of clustering. These properties are ubiquitous in many real-world networks and make model development and analysis much more challenging. To this end, I have derived and analysed a compact pairwise model with the number of equations being independent of the range of node degrees, and investigated the effects of clustering on epidemic dynamics. My thesis culminated with the third part where I explored the relationships between several different modelling methodologies, and derived an original non-Markovian Edge-Based Compartmental Model (EBCM) which allows both transmission and recovery to be arbitrary independent stochastic processes. The major result is a rigorous mathematical proof that the message passing (MP) model and the EBCM are equivalent, and thus, the EBCM is statistically exact on the ensemble of configuration model networks. From this consideration I derived a generalised pairwise-like model which I then used to build a model hierarchy, and to show that, given corresponding parameters and initial conditions, these models are identical to MP model or EBCM. In the final part of my thesis I considered the important problem of coupling epidemic dynamics with changes in network structure in response to the perceived risk of the epidemic. This was framed as a susceptible-infected-susceptible (SIS) model on an adaptive network, where susceptible nodes can disconnect from infected neighbours and, after some fixed time delay, connect to a random susceptible node that they are not yet connected to. This model assumes that nodes have perfect information on the state of all other nodes. Robust oscillations were found in a significant region of the parameter space, including an enclosed region known as an 'endemic bubble'. The major contribution of this work was to show that oscillations can occur in a wide region of the parameter space, this is in stark contrast with most previous research where oscillations were limited to a very narrow region of the parameter space. Any mathematical model is a simplification of reality where assumptions must be made. The models presented here show the importance of interrogating these assumptions to ensure that they are as realistic as possible while still being amenable to analysis.

The purpose of this PhD thesis is to study mesoscopic and macroscopic fluctuations in Interacting Particle Systems. The thesis is split into two main parts. In the first part, we consider a system of Ising spins interacting via Kac potential evolving with Glauber dynamics and study the macroscopic motion of an one-dimensional interface under forced displacement as the result of large scale fluctuations. In the second part, we consider a diffusive system modelled by a Simple Symmetric Exclusion Process (SSEP) which is driven out of equilibrium by the action of current reservoirs at the boundary and study the non-equilibrium fluctuations around the hydrodynamic limit for the SSEP with current reservoirs. We give a brief summary of the first part. In recent years, there has been significant effort to derive deterministic models describing two-phase materials and their dynamical properties. In this context, we investigate the law that governs the power needed to force a motion of a one dimensional macroscopic interface between two different phases of a given ferromagnetic sample with a prescribed speed V at low temperature. We show that given the mesoscopic deterministic non-local evolution equation for the magnetisation (a non local version of the Allen-Cahn equation), we consider a stochastic Ising spin system with Glauber dynamics and Kac interaction (the underlying microscopic stochastic process) whose mesoscopic scaling limit (intermediate scale between microscale and macroscale) is the given PDE, and we calculate the corresponding large deviations functional which would provide the action functional. We obtain that by deriving upper and lower bounds of the large deviation cost functional. Concepts from statistical mechanics such as contours, free energy, local equilibrium allow a better understanding of the structure of the cost functional. Then we characterise the limiting behaviour of the action functional under a parabolic rescaling, by proving that for small values of the ratio between the distance and the time, the interface moves with a constant speed, while for larger values the occurrence of nucleations is the preferred way to make the transition. This led to a production of two published papers [12] and [14] with my supervisor D. Tsagkarogiannis and N. Dirr. In the second part we study the non-equilibrium fluctuations of a system modelled by SSEP with current reservoirs around its hydrodynamic limit. In particular, we prove that, in the limit, the appropriately scaled fluctuation field is given by a Generalised Ornstein- Uhlenbeck process. For the characterisation of the limiting fluctuation field we implement the Holley-Stroock theory. This is not straightforward due to the boundary terms coming from the nature of the model. Hence, by following a martingale approach (martingale decomposition) and the derivation of the equation of the variance for this model combined with “good” enough correlation estimates (the so-called v-estimates), we reduce the problem to a form whose Holley-Stroock result in [45] is now applicable. This is work in progress jointly with my supervisor and P. Gonçalves, [13].