This thesis focusses on the properties of, and relationships between, several fundamental objects arising from critical physical models. In particular, we consider Schramm--Loewner evolutions, the Gaussian free field, Liouville quantum gravity and the Brownian continuum random tree. We begin by considering branching diffusions in a bounded domain $D\subset$ $R^{d}$, in which particles are killed upon hitting the boundary $\partial D$. It is known that such a system displays a phase transition in the branching rate: if it exceeds a critical value, the population will no longer become extinct almost surely. We prove that at criticality, under mild assumptions on the branching mechanism and diffusion, the genealogical tree associated with the process will converge to the Brownian CRT. Next, we move on to study Gaussian multiplicative chaos. This is the rigorous framework that allows one to make sense of random measures built from rough Gaussian fields, and again there is a parameter associated with the model in which a phase transition occurs. We prove a uniqueness and convergence result for approximations to these measures at criticality. From this point onwards we restrict our attention to two-dimensional models. First, we give an alternative, ``non-Gaussian" construction of Liouville quantum gravity (a special case of Gaussian multiplicative chaos associated with the 2-dimensional Gaussian free field), that is motivated by the theory of multiplicative cascades. We prove that the Liouville (GMC) measures associated with the Gaussian free field can be approximated using certain sequences of ``local sets" of the field. This is a particularly natural construction as it is both local and conformally invariant. It includes the case of nested CLE$_{4}$, when it is coupled with the GFF as its set of ``level lines". Finally, we consider this level line coupling more closely, now when it is between SLE$_{4}$ and the GFF. We prove that level lines can be defined for the GFF with a wide range of boundary conditions, and are given by SLE$_{4}$-type curves. As a consequence, we extend the definition of SLE$_{4}(\rho)$ to the case of a continuum of force points.

This thesis studies numerical methods for integrating the master equations describing Markov chain models of cardiac ion channels. Such models describe the time evolution of the probability that ion channels are in a particular state. Numerical simulations of such models are often computationally demanding because many solvers require relatively small time steps to ensure numerical stability. The aim of this project is to analyse selected Markov chains and develop more efficient and accurate solvers. We separate a Markov chain model into fast and slow time-scales based on the speed of transitions between states. Eliminating the fast transitions, we find an asymptotic reduction of zeroth-order and first-order in a small parameter describing the time-scales separation. We apply the theory to a Markov chain model of the fast sodium channel INa. We consider several variants for classifying some transitions as fast in order to find reduced systems that yield a good accuracy. However, the time step size is still restricted by numerical instabilities. We adapt the Rush-Larsen technique originally developed for gate models. Assuming that a transition matrix can be considered constant during each time step, we solve the Markov chain model analytically. The solution provides a recipe for a stable exponential solver, which we call "Matrix Rush-Larsen" (MRL). Using operator splitting we design an even more flexible "hybrid" method that combines the MRL with other solvers. The resulting improvement in stability allows a large increase in the time step size. In some models, we obtain reasonably accurate results 27 times faster using a hybrid method than with the forward Euler method, even with the maximal time step allowed by the stability constraint. Finally, we extend the cardiac simulation package BeatBox by the developed exponential solvers. We upgrade a format of "ionic" modules which describe a cardiac cell, in order to allow for a specific definition of Markov chain models. We also modify a particular integrator for ionic modules to include the MRL and the hybrid method. To test the functionality of the code, we have converted a number of cellular models into the ionic format. The documented code is available in the official BeatBox package distribution.

We consider linear and semilinear stochastic partial differential equations that in some sense can be viewed as being at the "endpoints" of the classical variational theory by Krylov and Rozovskii [25]. In terms of regularity of the coeffcients, the minimal assumption is boundedness and measurability, and a unique L2- valued solution is then readily available. We investigate its further properties, such as higher order integrability, boundedness, and continuity. The other class of equations considered here are the ones whose leading operators do not satisfy the strong coercivity condition, but only a degenerate version of it, and therefore are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes.

This thesis develops two variant statistical models for count data based upon compound models for contexts when the counts may be viewed as derived from two generations, which may or may not be independent. Unlike standard compound models, the variants model the sum of both generations. We consider cases where both generations are negative binomial or one is Poisson and the other is negative binomial. The first variant, denoted SVA, follows a zero restriction, where a zero in the first generation will automatically be followed by a zero in the second generation. The second variant, denoted SVB, is a convolution model that does not possess this zero restriction. The main properties of the SVA and SVB models are outlined and compared with standard compound models. The results show that the SVA distributions are similar to standard compound distributions for some fixed parameters. Comparisons of SVA, Poisson hurdle, negative binomial hurdle and their zero-inflated counterpart using simulated SVA data indicate that different models can give similar results, as the generating models are not always selected as the best fitting. This thesis focuses on the use of the variant models to model citation counts. We show that the SVA models are more suitable for modelling citation data than other previously used models such as the negative binomial model. Moreover, the application of SVA and SVB models may be used to describe the citation process. This thesis also explores model selection techniques based on log-likelihood methods, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The suitability of the models is also assessed using two diagrammatic methods, randomised quantile residual plots and Christmas tree plots. The Christmas tree plots clearly illustrate whether the observed data are within fluctuation bounds under the fitted model, but the randomised quantile residual plots utilise the cumulative distribution, and hence are insensitive to individual data values. Both plots show the presence of citation counts that are larger than expected under the fitted model in the data sets.

In this thesis we study three different modelling frameworks for biological systems of dispersal and combinations thereof. The three frameworks involved are individual-based models, group-level models in the form of partial differential equations (PDEs) and robot swarms. In the first two chapters of the thesis, we present ways of coupling individual based models with PDEs in so-called hybrid models, with the aim of achieving improved performance of simulations. Two classes of such hybrid models are discussed that allow an efficient simulation of multi-species systems of dispersal with reactions, but involve individual resolution for certain species and in certain parts of a computational domain if desired. We generally consider two types of example systems: bacterial chemotaxis and reaction-diffusion systems, and present results in the respective application area as well as general methods. The third chapter of this thesis introduces swarm robotic experiments as an additional tool to study systems of dispersal. In general, those experiments can be used to mimic animal behaviour and to study the impact of local interactions on the group-level dynamics. We concentrate on a target finding problem for groups of robots. We present how PDE descriptions can be adjusted to incorporate the finite turning times observed in the robotic system and that the adjusted models match well with experimental data. In the fourth and last chapter, we consider interactions between robots in the form of hard-sphere collisions and again derive adjusted PDE descriptions. We show that collisions have a significant impact on the speed with which the group spreads across a domain. Throughout these two chapters, we apply a combination of experiments, individual-based simulations and PDE descriptions to improve our understanding of interactions in systems of dispersal.

Periodically correlated time series generally exist in several fields including hydrology, climatology, economics and finance, and are commonly modelled using periodic autoregressive (PAR) model. For a time series with stochastic periodic trend, for which a unit root is expected, a periodically integrated autoregressive PIAR model with periodic and/or seasonal unit root has been shown to be a satisfactory model. The existing theory used the multivariate methodology to study PIAR models. However, this theory is convoluted, majority of it only developed for quarterly time series and its generalisation to time series with larger number of periods is quite cumbersome. This thesis studies the existing theory and highlights its restrictions and flaws. It provides a coherent presentation of the steps for analysing PAR and PIAR models for different number of periods. It presents the different unit roots representations and compares the performance of different unit root tests available in literature. The restrictions of existing studies gave us the impetus to develop a unified theory that gives a clear understanding of the integration and unit roots in the periodic models. This theory is based on the spectral information of the multi-companion matrix of the periodic models. It is more general than the existing theory, since it can be applied to any number of periods whereas the existing methods are developed for quarterly time series. Using the multi-companion method, we specify and estimate the periodic models without the need to extract complicated restrictions on the model parameters corresponding to the unit roots, as required by NLS method. The multi-companion estimation method performed well and its performance is equivalent to the NLS estimation method that has been used in the literature. Analysing integrated multivariate models is a problematic issue in time series. The multi-companion theory provides a more general approach than the error correction method that is commonly used to analyse such time series. A modified state state representation for the seasonal periodically integrated autoregressive (SPIAR) model with periodic and seasonal unit roots is presented. Also an alternative state space representations from which the state space representations of PAR, PIAR and the seasonal periodic autoregressive (SPAR) models can be directly obtained is proposed. The seasons of the parameters in these representations have been clearly specified, which guarantees correct estimated parameters. Kalman filter have been used to estimate the parameters of these models and better estimation results are obtained when the initial values were estimated rather than when they were given.

In recent times, credit risk analysis has grown to become one of the most important problems dealt with in the mathematical finance literature. Fundamentally, the problem deals with estimating the probability that an obligor defaults on their debt in a certain time. To obtain such a probability, several methods have been developed which are regulated by the Basel Accord. This establishes a legal framework for dealing with credit and market risks, and empowers banks to perform their own methodologies according to their interests under certain criteria. Credit risk analysis is founded on the rating system, which is an assessment of the capability of an obligor to make its payments in full and on time, in order to estimate risks and make the investor decisions easier.Credit risk models can be classified into several different categories. In structural form models (SFM), that are founded on the Black & Scholes theory for option pricing and the Merton model, it is assumed that default occurs if a firm's market value is lower than a threshold, most often its liabilities. The problem is that this is clearly is an unrealistic assumption. The factors models (FM) attempt to predict the random default time by assuming a hazard rate based on latent exogenous and endogenous variables. Reduced form models (RFM) mainly focus on the accuracy of the probability of default (PD), to such an extent that it is given more importance than an intuitive economical interpretation. Portfolio reduced form models (PRFM) belong to the RFM family, and were developed to overcome the SFM's difficulties.Most of these models are based on the assumption of having an underlying Markovian process, either in discrete or continuous time. For a discrete process, the main information is containted in a transition matrix, from which we obtain migration probabilities. However, according to previous analysis, it has been found that this approach contains embedding problems. The continuous time Markov process (CTMP) has its main information contained in a matrix Q of constant instantaneous transition rates between states. Both approaches assume that the future depends only on the present, though previous empirical analysis has proved that the probability of changing rating depends on the time a firm maintains the same rating. In order to face this difficulty we approach the PD with the continuous time semi-Markov process (CTSMP), which relaxes the exponential waiting time distribution assumption of the Markovian analogue.In this work we have relaxed the constant transition rate assumption and assumed that it depends on the residence time, thus we have derived CTSMP forward integral and differential equations respectively and the corresponding equations for the particular cases of exponential, gamma and power law waiting time distributions, we have also obtained a numerical solution of the migration probability by the Monte Carlo Method and compared the results with the Markovian models in discrete and continuous time respectively, and the discrete time semi-Markov process. We have focused on firms from U.S.A. and Canada classified as financial sector according to Global Industry Classification Standard and we have concluded that the gamma and Weibull distribution are the best adjustment models.

Cette thèse étudie des modèles de populations branchantes appelés arbres de ramification, dans lesquels les individus évoluent indépendamment les uns des autres, ont des durées de vie indépendantes, identiquement distribuées (non nécessairement exponentielles), et donnent naissance à taux constant au cours de leur vie. On enrichit ces modèles en supposant que chaque individu porte un type et peut subir à la naissance une mutation, qui lui confère un nouveau type. On démontre dans le premier chapitre des résultats théoriques de convergence en loi pour des processus de Lévy bivariés sans sauts négatifs. Ces résultats sont ensuite exploités dans le deuxième chapitre pour établir un principe d'invariance pour l'arbre généalogique des populations décrites ci-dessus, enrichi de leur historique mutationnel, dans une asymptotique de grande taille de population. Enfin, on étudie dans le troisième chapitre la structure généalogique et le spectre de fréquence par site (nombre de mutations portées par un nombre donné d'individus) d'échantillons uniformes dans des populations branchantes critiques dont la limite d'échelle est un arbre brownien (par exemple, des arbres de naissance et mort critiques). Des perspectives d'applications de ces résultats à la génétique des populations sont présentées dans le quatrième chapitre. / This thesis studies branching population models called splitting trees, where individuals evolve independently from one another, have independent and identically distributed lifetimes (that are not necessarily exponential), and give birth at constant rate during their lives. We further assume that each individual carries a type, and possibly undergoes a mutation at her birth, that changes her type into a new one. In the first chapter, we prove convegence results for bivariate Lévy processes with non negative jumps. These theoretical results are used in the second chapter to establish an invariance principle for the genealogical tree of the populations described above, enriched with their mutational history, in a large population size asymptotic. Finally we study in the third chapter the genealogical structure and the site frequency spectrum (number of mutations carried by a given number of individuals) for uniform samples in critical branching populations whose scaling limit is a Brownian tree (e.g., critical birth-death trees). Possible future applications of these results to population genetics are presented in the fourth chapter.

Cette thèse est consacrée à l'étude asymptotique d'objets liés au mouvement brownien sur le groupe unitaire en grande dimension, ainsi qu'à l'étude, dans le cadre des probabilités libres, des versions non-commutatives de ces objets. Elle se subdivise essentiellement en trois parties.Dans le chapitre 2, nous résolvons le problème initial de cette thèse, à savoir la convergence de la transformation de Hall sur le groupe unitaire vers la transformation de Hall libre, lorsque la dimension tend vers l'infini. Pour résoudre ce problème, nous établissons des théorèmes d'existence de noyaux de transition pour la convolution libre. Enfin, nous utilisons ces résultats pour prouver que, pareillement au mouvement brownien sur le groupe unitaire, le mouvement brownien sur le groupe linéaire converge en distribution non-commutative vers sa version libre. Nous étudions les fluctuations autour de cette convergence dans le chapitre 3. Le chapitre 4 présente un morphisme entre les mesures infiniment divisibles pour la convolution libre additive d'une part et multiplicative de l'autre. Nous montrons que ce morphisme possède une version matricielle qui s'appuie sur un nouveau modèle de matrices aléatoires pour les processus de Lévy libres multiplicatifs. / This thesis focuses on the asymptotic of objects related to the Brownian motion on the unitary group in large dimension, and on the study, in free probability, of the non-commutative versions of those objects. It subdivides into essentially three parts.In Chapter 2, we solve the original problem of this thesis: the convergence of the Hall transform on the unitary group to the free Hall transform, as the dimension tends to infinity. To solve this problem, we establish theorems of existence of transition kernel for the free convolution. Finally, we use these results to prove that, exactly as the Brownian motion on the unitary group, the Brownian motion on the linear group converges in noncommutative distribution to its free version. Then we study the fluctuations around this convergence in Chapter 3. Chapter 4 presents a homomorphism between infinitely divisible measures for the free convolution, in respectively the additive case and the multiplicative case. We show that this homomorphism has a matricialversion which is based on a new model of random matrices for the free multiplicative Lévy processes.

Cette thèse est consacrée à l'étude de décompositions trajectorielles de processus de Lévy spectralement positifs et des relations de dualité pour des processus de ramification, motivée par l'utilisation de ces derniers comme modèles probabilistes d'une dynamique épidémiologique. Nous modélisons l'arbre de transmission d'une maladie comme un arbre de ramification, où les individus évoluent indépendamment les uns des autres, ont des durées de vie i.i.d. (périodes d'infectiosité) et donnent naissance (infections secondaires) à un taux constant durant leur vie. Le processus d'incidence dans ce modèle est un processus de Crump-Mode-Jagers (CMJ) et le but principal des deux premiers chapitres est d'en caractériser la loi conjointement avec l'arbre de transmission partiellement observé, inferé à partir des données de séquences. Dans le Chapitre I, nous obtenons une description en termes de fonctions génératrices de la loi du nombre d'individus infectieux, conditionnellement à l'arbre de transmission reliant les individus actuellement infectés. Une version plus élégante de cette caractérisation est donnée dans le Chapitre II, en passant par un résultat général d'invariance par retournement du temps pour une classe de processus de ramification. Finallement, dans le Chapitre III nous nous intéressons à la loi d'un processus de ramification (sous)critique vu depuis son temps d'extinction. Nous obtenons un résultat de dualité qui implique en particulier l'invariance par retournement du temps depuis leur temps d'extinction des processus CMJ (sous)critiques et de l'excursion hors de 0 de la diffusion de Feller critique (le processus de largeur de l'arbre aléatoire de continuum). / This dissertation is devoted to the study of some pathwise decompositions of spectrally positive Lévy processes, and duality relationships for certain (possibly non-Markovian) branching processes, driven by the use of the latter as probabilistic models of epidemiological dynamics. More precisely, we model the transmission tree of a disease as a splitting tree, i.e. individuals evolve independently from one another, have i.i.d. lifetimes (periods of infectiousness) that are not necessarily exponential, and give birth (secondary infections) at a constant rate during their lifetime. The incidence of the disease under this model is a Crump-Mode-Jagers process (CMJ); the overarching goal of the two first chapters is to characterize the law of this incidence process through time, jointly with the partially observed (inferred from sequence data) transmission tree. In Chapter I we obtain a description, in terms of probability generating functions, of the conditional likelihood of the number of infectious individuals at multiple times, given the transmission network linking individuals that are currently infected. In the second chapter, a more elegant version of this characterization is given, passing by a general result of invariance under time reversal for a class of branching processes. Finally, in Chapter III we are interested in the law of the (sub)critical branching process seen from its extinction time. We obtain a duality result that implies in particular the invariance under time reversal from their extinction time of the (sub)critical CMJ processes and the excursion away from 0 of the critical Feller diffusion (the width process of the continuum random tree).