Les cartes planaires sont des graphes planaires dessinés sur la sphère et vus à déformation près. De nombreuses propriétés des cartes sont supposées universelles, dans le sens où elles ne dépendent pas des détails du modèle choisi. Nous commençons par établir une inégalité isopérimétrique dans la quadrangulation infinie du plan. Nous confirmons également une conjecture de Krikun portant sur la longueur des cycles les plus courts séparant la boule de rayon $r$ de l'infini. Dans un deuxième temps, nous nous intéressons à l'effet de modifications de distances sur la géométrie à grande échelle des quadrangulations uniformes, élargissant la classe d'universalité de la carte brownienne. Nous montrons également que la bijection de Tutte, entre quadrangulations et cartes planaires, est asymptotiquement une isométrie. Enfin, nous établissons une borne supérieure sur le temps de mélange de la marche aléatoire dans les cartes aléatoires. / Planar maps are planar graphs drawn on the sphere and seen up to deformation. Many properties of maps are conjectured to be universal, in the sense that they do not depend on the details of the model.We begin by establishing an isoperimetric inequality in the infinite quadrangulation of the plane. We also confirm a conjecture by Krikun concerning the length of the shortest cycles separating the ball of radius $r$ from infinity. We then consider the effect of modifications of distances on the large-scale geometry of uniform quadrangulations, extending the universality class of the Brownian map. We also show that the Tutte bijection, between quadrangulations and planar maps, is asymptotically an isometry. Finally, we establish an upper bound on the mixing time of the random walk in random maps.

In this thesis, we study various notions of variation of certain stochastic processes, namely $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. We study these concepts for fractional Brownian motions and Rosenblatt processes. A fractional Brownian motion is a Gaussian process and it has been intensively developed and studied over the last two decades because of its importance in modeling various phenomena. On the other hand, a Rosenblatt process, which is a non- Gaussian process that can be used for modeling non-Gaussian fluctuations, has not been getting as much attention as fractional Brownian motion. For that reason, we concentrate in this thesis on this process and we present some original results that deal with ergodicity, $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. Boris Kiška

Parametric models are used to understand dynamical systems and predict its future behavior. It is difficult to estimate the model’s parametric values since there are usually many parameters and they are highly correlated. The aim of this project is to apply the method of indirect inference and the two stage approach to estimate the drift and volatility parameters of a Geometric Brownian Motion. This was first done by estimating the parameters of a known Geometric Brownian process. Then, the Coca-Cola Company’s stock was used for a five-year forecast to study the estimators’ predictive power. The two stage approach struggles when the data does not truly follow a Geometric Brownian Motion, but when it does it produces highly efficient and accurate estimates. The method of indirect inference produces better estimates, than the two stage approach,for data that deviates from a Geometric Brownian Motion.Therefore, it is preferable to use indirect inference over two stage approach for stock price forecasting. / Parametriska modeller används för attförstå dynamiska system och förutspå dess framtida beteende.Det är utmanande att skatta modellens parametriska värdeneftersom det vanligtvis finns många parametrar och de är oftastarkt korrelerade. Målet med detta projekt är att tillämpametoderna indirect inference och two stage approach för attskatta drivnings- och volatilitetsparametrarna av en geometriskBrownsk rörelse. Först skattades parametrarna av en kändGeometrisk Brownsk rörelse. Sedan användes The Coca-ColaCompanys aktie i syfte att studera estimatorernas förmåga attförutspå en femårig period. Two stage approach fungerar dåligtför data som inte helt följer en geometrisk Brownsk rörelse, mennär datan gör det är skattningarna noggranna och effektiva.Indirect inference ger bättre skattningar än two stage approachnär datan inte helt följer en geometrisk Brownsk rörelse. Därförär indirect inference att föredra för aktieprognoser. / Kandidatexjobb i elektroteknik 2022, KTH, Stockholm

The aim of this thesis is to simulate stochastic models that are driven by a fractional Brownian motion process and to apply these methods to financial applications related to yield rate and asset price modelling. Several rough volatility processes are used to model the asset price and yield dynamics. Firstly fractional processes of Cox-Ingersoll-Ross, CEV and Vasicek types are introduced as models for volatility and yield data. In this framework it holds that the Hurst parameter that determines the covariance structure of the fBM process can be directly estimated from observed data series using a least squares log-periodogram approach. The remaining parameters in the model are estimated using a combination of Maximum Likelihood estimates and expectation estimations. In the modelling and pricing of assets one model that is studied is the fractional Heston model, that is used to model an asset price process using both observed asset and volatility data. Similarly two other similar rough volatility models are also studied, which are constructed so as to have log-Normal returns. These processes which in the thesis are called the exponential models 1 and 2 have rough volatility that are characterized by the CEV and Vasicek processes. Additionally the first order Wiener Chaos Expansion is implemented and explored in two ways. Firstly the Chaos Expansion is applied to a parametric fractional stochastic model which is used to generate a Wick product process, which is found to resemble the underlying process. It is also used to generate an approximate expansion of real yield rate data using a bootstrap sampling approach. / Den här uppsatsen syftar till att simulera stokastiska modeller som drivs av fraktionell Brownsk rörelse och att använda dessa modeller i finansiella tillämpningar relaterade till räntor och finansiella tillgångar. Flera volatilitetsprocesser som är rough används för att modellera ränte- och aktiedynamiken. Först introduceras de fraktionella varianterna av Cox-Ingersoll-Ross, CEV och Vasicek processer, vilka används för att modellera volatilitet och ränteprocesser. Med detta tillvägagångssätt gäller det att Hurstparametern, vilken bestämmer covariansstrukturen för den fraktionella Brownska rörelsen, kan uppskattas direkt från observerad data med en minsta kvadrat log-periodogram-metod. Samtliga andra parametrar i modellen uppskattas med en kombination av Maximum Likelihood och uppskattning av väntevärden. I modelleringen och prissättningen av finansiella tillgångar är en model som studeras den fraktionella Hestonmodellen, som används för att modellera en tillgång baserat på både volatilitets- och aktiedata. Ytterligare två liknande modeller studeras, vilka också har volatilitet som är rough och är konstruerade så att deras avkastning är log-Normal. Dessa processer, vilka i uppsatsen är benämnda som de exponentiella modellerna 1 och 2 har volatilitet som karaktäriseras av CEV- och Vasicekprocesser. Ytterligare är Wiener's Kaosexpansion av första ordningen också implementerad och undersöks från två håll. Först används den på en parameterbestämd fraktionell stokastisk modell, vilken används för att generera en Wickproduktprocess. Expansionen används även med hjälp av en bootstrap-metod för att generera en process från observerad data.

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In this thesis, we study the kinetic disordering of systems interacting with an agent or a walker. Our studies divide naturally into two classes: for the first, the dynamics of the walker conserves the total magnetization of the system, for the second, it does not. These distinct dynamics are investigated in part I and II respectively. In part I, we investigate the disordering of an initially phase-segregated binary alloy due to a highly mobile vacancy which exchanges with the alloy atoms. This dynamics clearly conserves the total magnetization. We distinguish three versions of dynamic rules for the vacancy motion, namely a pure random walk , an "active" and a biased walk. For the random walk case, we review and reproduce earlier work by Z. Toroczkai et. al., [9] which will serve as our base-line. To test the robustness of these findings and to make our model more accessible to experimental studies, we investigated the effects of finite temperatures ("active walks") as well as external fields (biased walks). To monitor the disordering process, we define a suitable disorder parameter, namely the number of broken bonds, which we study as a function of time, system size and vacancy number. Using Monte Carlo simulations and a coarse-grained field theory, we observe that the disordering process exhibits three well separated temporal regimes. We show that the later stages exhibit dynamic scaling, characterized by a set of exponents and scaling functions. For the random and the biased case, these exponents and scaling functions are computed analytically in excellent agreement with the simulation results. The exponents are remarkably universal. We conclude this part with some comments on the early stage, the interfacial roughness and other related features. In part II, we introduce a model of binary data corruption induced by a Brownian agent or random walker. Here, the magnetization is not conserved, being related to the density of corrupted bits ρ. Using both continuum theory and computer simulations, we study the average density of corrupted bits, and the associated density-density correlation function, as well as several other related quantities. In the second half, we extend our investigations in three main directions which allow us to make closer contact with real binary systems. These are i) a detailed analysis of two dimensions, ii) the case of competing agents, and iii) the cases of asymmetric and quenched random couplings. Our analytic results are in good agreement with simulation results. The remarkable finding of this study is the robustness of the phenomenological model which provides us with the tool, continuum theory, to understand the nature of such a simple model. / Ph. D.

[pt] A escolha da cidade do Rio de Janeiro como sede da Olimpíada de 2016 trouxe a necessidade de realização de diversos projetos de infraestrutura de transportes. Um destes projetos envolveu a revitalização da zona portuária, conhecido como Projeto Porto Maravilha, e entre as melhorias projetadas estava a implantação de um novo modal de transportes sobre trilhos, o VLT Carioca. Este trabalho analisa o projeto em regime de parceria público-privada do VLT Carioca na zona portuária da cidade do Rio de Janeiro através da teoria de opções reais. O objetivo do estudo é determinar o retorno esperado do projeto para o consórcio vencedor da licitação, analisar o impacto dos incentivos governamentais para o parceiro privado e os custos totais do projeto para o Estado. A demanda estocástica é modelada por meio do movimento geométrico browniano (MGB), e os resultados indicam que o projeto tem um retorno relativamente pequeno em relação ao investimento inicial, as garantias oferecidas pela Prefeitura aumentam o valor do projeto e a realização do projeto sob a modalidade de parceria público-privada traz para o parceiro público uma economia de aproximadamente 50 por cento do valor total. / [en] The choice of the city of Rio de Janeiro to must the 2016 Olympics games brought the need to carry out transportation infrastructure projects. One of these projects involved the revitalization of the port area, known as the Porto Maravilha Project. One of the improvements projected was a new modal rail transport, the VLT Carioca. This paper analyzes the public-private partnership project VLT Carioca in port area of Rio de Janeiro city using real options. The purpose of this study is to determine the expected return of project for winning bidding consortium, analyzing the impact of government incentives to private partner, and the total costs to state. Stochastic demand is modeled as a Brownian geometric motion (GBM). The results indicate that the project has a small return on the initial investment, the guarantees offered by government increase the value of the project and the realization of the project under the public-private partnership modality brings to the public partner a gain of economy approximately 50 percent of the value.

L’objet de cette thèse est d’étudier plusieurs modèles probabilistes reliant les marches aléatoires et les arbres aléatoires issus de processus de branchement critiques.Dans la première partie, nous nous intéressons au modèle de marche aléatoire à valeurs dans un réseau euclidien et indexée par un arbre de Galton–Watson critique conditionné par la taille. Sous certaines hypothèses sur la loi de reproduction critique et la loi de saut centrée, nous obtenons, dans toutes les dimensions, la vitesse de croissance asymptotique du nombre de points visités par cette marche, lorsque la taille de l’arbre tend vers l’infini. Ces résultats nous permettent aussi de décrire le comportement asymptotique du nombre de points visités par une marche aléatoire branchante, quand la taille de la population initiale tend vers l’infini. Nous traitons également en parallèle certains cas où la marche aléatoire possède une dérive constante non nulle.Dans la deuxième partie, nous nous concentrons sur les propriétés fractales de la mesure harmonique des grands arbres de Galton–Watson critiques. On comprend par mesure harmonique la distribution de sortie, hors d’une boule centrée à la racine de l’arbre, d’une marche aléatoire simple sur cet arbre. Lorsque la loi de reproduction critique appartient au domaine d’attraction d’une loi stable, nous prouvons que la masse de la mesure harmonique est asymptotiquement concentrée sur une partie de la frontière, cette partie ayant une taille négligeable par rapport à celle de la frontière. En supposant que la loi de reproduction critique a une variance finie, nous arrivons à évaluer la masse de la mesure harmonique portée par un sommet de la frontière choisi uniformément au hasard. / The aim of this Ph. D. thesis is to study several probabilistic models linking the random walks and the random trees arising from critical branching processes.In the first part, we consider the model of random walk taking values in a Euclidean lattice and indexed by a critical Galton–Watson tree conditioned by the total progeny. Under some assumptions on the critical offspring distribution and the centered jump distribution, we obtain, in all dimensions, the asymptotic growth rate of the range of this random walk, when the size of the tree tends to infinity. These results also allow us to describe the asymptotic behavior of the range of a branching random walk, when the size of the initial population goes to infinity. In parallel, we treat likewise some cases where the random walk has a non-zero constant drift.In the second part, we focus on the fractal properties of the harmonic measure on large critical Galton–Watson trees. By harmonic measure, we mean the exit distribution from a ball centered at the root of the tree by simple random walk on this tree. If the critical offspring distribution is in the domain of attraction of a stable distribution, we prove that the mass of the harmonic measure is asymptotically concentrated on a boundary subset of negligible size with respect to that of the boundary. Assuming that the critical offspring distribution has a finite variance, we are able to calculate the mass of the harmonic measure carried by a random vertex uniformly chosen from the boundary.

Cette thèse est consacrée à l’étude des équations aux dérivées partielles stochastiques de type parabolique. Dans la première partie nous démontrons de nouveaux résultats concernant l’existence et l’unicité de solutions variationnelles globales et locales à des problèmes avec des conditions aux bords de type Neumann pour une classe d’équations aux dérivées partielles stochastiques non-autonomes. Les équations que nous considérons sont définies sur des domaines non bornés de l’espace euclidien qui satisfont à certaines conditions géométriques, et sont dirigées par un bruit multiplicatif dérivé d’un processus de Wiener fractionnaire infini-dimensionnel caractérisé par une suite de paramètres de Hurst H = (Hi) i ∈ N+ ⊂ (1/2,1). Ces paramètres sont en fait soumis à d’autres contraintes intimement liées à la nature de la non-linéarité dans le terme stochastique des équations, et au choix des espaces fonctionnels dans lesquels le problème à résoudre est bien posé. Notre méthode de preuve repose essentiellement sur des arguments d’injections compactes. Dans la seconde partie, nous étudions la possibilité de l’explosion de solutions d’une classe d’équations aux dérivées partielles stochastiques semi-linéaire avec des conditions aux bords de type Dirichlet, perturbées par un mélange d’un mouvement brownien et d’un mouvement brownien fractionnaire et dirigées par une classe d’opérateurs différentiels non autonomes contenant des processus de diffusions et des processus de Lévy. Notre but est de comprendre l’influence de la partie stochastique et de l’opérateur différentiel sur le comportement d’explosion des solutions. En particulier, nous donnons des expressions explicites pour des bornes inférieures et supérieures du temps de l’explosion de la solution, et des conditions suffisantes pour l’existence d’une solution globale positive. Nous estimons également la probabilité d’une explosion en temps fini et la loi d’une borne supérieur du temps d’explosion de la solution / This thesis is concerned with stochastic partial differential equations of parabolic type. In the first part we prove new results regarding the existence and the uniqueness of global and local variational solutions to a Neumann initial-boundary value problem for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H = (Hi) i ∈ N+ ⊂ (1/2,1). These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way. The second part is devoted to the study of the blowup behavior of solutions to semilinear stochastic partial differential equations with Dirichlet boundary conditions driven by a class of differential operators including (not necessarily symmetric) Lévy processes and diffusion processes, and perturbed by a mixture of Brownian and fractional Brownian motions. Our aim is to understand the influence of the stochastic part and that of the differential operator on the blowup behavior of the solutions. In particular we derive explicit expressions for an upper and a lower bound of the blowup time of the solution and provide a sufficient condition for the existence of global positive solutions. Furthermore, we give estimates of the probability of finite time blowup and for the tail probabilities of an upper bound for the blowup time of the solutions

The work consists of two parts. In the first part which is concerned with random walks, we construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob h-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an h-transform in the Weyl chamber of type C. In the second part which is concerned with Brownian motion, we examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.