Electronic energy transfer has been investigated in pure donor systems by means of computer simulations. Calculated properties were the probability that the initially excited donor is excited at a time t after the excitation, Gs(t), the mean square displacement of the excitation and different fluorescence observables. For three dimensional systems the results obtained by Monte Carlo simulations were compared to the so-called GAF-theory {Gouchanour,C. R., Andersen, H. C. and Fayer, M. D., J. Chem. Phys. 81, 4380 (1984)}, and the agreement was found to be good. Anisotropic systems, i.e. mono-, bi- and multilayer systems, were compared to the two-particle model {Baumann,J. and Fayer, M. D., J. Chem. Phys. 85, 4087 (1986)}. The agreement between the Gs(t) calculated from the tp- model and the Monte Carlo simulations were good for all systems investigated. However, the agreement between the fluorescence observables obtained by MC and the tp-model were in general poor. A much better agreement was found when a phenomenological approach was used for calculating the fluorescence depolarization ratios. Three dimensional systems where the donors are rotating on the same time scale as the energy transfer takes place have also been studied and compared to analytical theories. The Molecular Dynamics simulations of decapeptide H142 shows that simulations in a continuum with a relative permeability do not provide a reliable alternative to simulations with explicit solvent molecules. / <p>Diss. (sammanfattning) Umeå : Umeå universitet, 1991, härtill 5 uppsatser</p> / digitalisering@umu

Stochastic models describing how interacting individuals give rise to collective behaviour have become a widely used tool across disciplines—ranging from biology to physics to social sciences. Continuum population-level models based on partial differential equations for the population density can be a very useful tool (when, for large systems, particle-based models become computationally intractable), but the challenge is to predict the correct macroscopic description of the key attributes at the particle level (such as interactions between individuals and evolution rules). In this thesis we consider the simple class of models consisting of diffusive particles with short-range interactions. It is relevant to many applications, such as colloidal systems and granular gases, and also for more complex systems such as diffusion through ion channels, biological cell populations and animal swarms. To derive the macroscopic model of such systems, previous studies have used ad hoc closure approximations, often generating errors. Instead, we provide a new systematic method based on matched asymptotic expansions to establish the link between the individual- and the population-level models. We begin by deriving the population-level model of a system of identical Brownian hard spheres. The result is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. We then expand this core problem in several directions. First, for a system with two types of particles (two species) we obtain a nonlinear cross-diffusion model. This model captures both alternative notions of diffusion, the collective diffusion and the self-diffusion, and can be used to study diffusion through obstacles. Second, we study the diffusion of finite-size particles through confined domains such as a narrow channel or a Hele–Shaw cell. In this case the macroscopic model depends on a confinement parameter and interpolates between severe confinement (e.g., a single- file diffusion in the narrow channel case) and an unconfined situation. Finally, the analysis for diffusive soft spheres, particles with soft-core repulsive potentials, yields an interaction-dependent non-linear term in the diffusion equation.

Dans ce travail, nous nous sommes intéressés d'une part à la théorie du chaos multiplicatif Gaussien introduite par Kahane en 1985 et d'autre part à la théorie des matrices aléatoires dont les pionniers sont Wigner, Wishart et Dyson. La première partie de ce manuscrit contient une brève introduction à ces deux théories ainsi que les contributions personnelles de ce manuscrit expliquées rapidement. Les parties suivantes contiennent les textes des articles publiés [1], [2], [3], [4], [5] et pré-publiés [6], [7], [8] sur ces résultats dans lesquels le lecteur pourra trouver des développements plus détaillés / In this thesis, we are interested on the one hand in the theory of Gaussian multiplicative chaos introduced by Kahane in 1985 and on the other hand in random matrix theory whose pioneers are Wigner, Wishart and Dyson. The first part of this manuscript constitutes a brief introduction to those two theories and also contains the personal contributions of this work rapidly explained. The following parts contain the texts of the published articles [1], [2], [3], [4], [5] and pre-prints [6], [7], [8] on those results where the reader can find more detailed developments

Cette étude vise à une meilleure compréhension des phénomènes rencontrés en filtration des aérosols nanométriques, c’est-à-dire inférieurs à 100 nm, neutres et/ou chargés. Pour ce faire, trois différents types de média ont été étudiés : des grilles, en acier et matières synthétiques, des filtres non tissés, en fibres de verre ou polymériques, et des lits granulaires, constitués de billes d’acier ou de zéolithe. Il ressort des résultats expérimentaux obtenus que quel que soit le média testé, l’efficacité de collecte des particules augmente lorsque le diamètre de l’aérosol diminue, et ce jusque 4 nm. Ceci entre en contradiction avec l’approche théorique dite du rebond thermique, développée par Wang et Kasper en 1991, selon laquelle l’efficacité de collecte serait susceptible de diminuer en-dessous de 10 nm. La vérification des calculs de Wang et Kasper permet d’expliquer cette incohérence, et montre, à partir de valeurs plus réalistes de l’énergie d’adhésion particule-fibre, que si le rebond thermique existe, celui-ci ne pourra se manifester qu’en-dessous de 1 nm, au mieux. Ainsi, les perméances expérimentales des différents médias testés ont pu être modélisées en tenant compte des mécanismes de collecte par diffusion et/ou par effets électrostatiques. Une étude originale sur les performances, dans le domaine nanométrique, de filtres en fibres de verre intentionnellement percés complète ce travail. Pour un même média fibreux, la perméance augmente avec le diamètre de perforation réalisée. Par ailleurs, pour une taille de perforation donnée, la perméance devient indépendante du diamètre des particules en-dessous d’une taille limite, fonction de la dimension de la perforation. Il a enfin été mis en évidence que la baisse d’efficacité est d’autant plus importante que la résistance à l’écoulement de l’air du filtre est importante. Un modèle semi-empirique, fondé sur la différenciation du flux d’aérosol traversant la fuite du flux traversant le matelas fibreux résiduel du filtre, permet de bien représenter ces états de fait / This study aims to better understand the mechanisms encountered in nanoparticles aerosol filtration, the particles being charged or not. Three different types of media were studied: stainless steel or synthetics wire screens, unwoven filters in glass or polymer fibres, and at last, granular beds made from steel or zeolite balls. Experimental results show that, whatever the media, collection efficiency increases as the particle diameter decreases down to 4 nm. This point conflicts with the so-called thermal rebound effect developed by Wang and Kasper in 1991, according to which collection efficiency could decrease below 10 nm. The checking of Wang and Kasper’s calculations enables to explain this discrepancy and shows from more probable particle-to-fibre adhesion energy values that if thermal rebound phenomenon exists, it would only be measurable below 1 nm. Then, experimental points can be modelled from both diffusion and electrostatic forces collection mechanisms. An investigation on the filtration behaviour of fibreglass filters in the nanometric domain when intentionally-pierced with calibrated needles completes the above-mentioned works. For a same media, penetration increases as the leak diameter does. On the other hand, for a given hole size, penetration becomes independent of the particle diameter below a critical scale, which is a function of the leak diameter. It was lastly shown that the efficiency of a pierced media decreases all the more that its air flow resistance is higher. A semi-empirical model based on the differentiation between the aerosol flow across the leak and the one through the residual fibrous bed of the filter enables to well represent these points

Dans la première partie de cette thèse, on présente des inégalités de concentration convexe pour des intégrales stochastiques. Ces résultats sont obtenus par calcul stochastique e tpar calcul de Malliavin forward/backward. On présente également des inégalités de déviation pour les exponentielles martingales à saut.Dans une deuxième partie on présente des théorèmes limites pour le conditionnement du mouvement brownien. / In the first part of this thesis, we present some convex concentration inequalities for stochastic integrals. These results are obtained by forward/backward stochastic calculus combined with Malliavin calculus. We also present deviation inequalities for exponentialjump-diffusion.In the second part, we present some limit theorems for the conditionning of Brownian motion.

The diploma thesis describes portfolio management with proportional transaction costs. The main aim is to describe using of shadow prices to find the optimal Markov policies keeping the proportion of the investor's wealth invested in the risky asset within the corresponding interval in order to maximize the long run geometric growth rate. On the real market, the investor must pay transaction costs when he buys/sells shares. In the diploma thesis we transform these prices into the shadow price; when trading in the shadow price there are no transaction costs. The solution itself is based on Itô formula and the martingal theory. The prices of shares are modeled as geometric Brownian motion. Powered by TCPDF (www.tcpdf.org)

In the presented thesis we study three methods of pricing European currency barrier options. With help of these methods we value selected barrier options with underlying asset EUR/CZK. In the first chapter we introduce the basic definitions from the world of financial derivatives and we describe our data. In the second chapter we deal with the classical model based on geometric Brownian motion of underlying asset and we prove a theorem of valuating Up-In-barrier option in this model. In the third chapter we introduce a model with stochastic volatility, the Heston model. We calibrate this model to market data and we use it to value our barrier options. In the last chapter we describe a jump diffusion model. Again we calibrate this jump diffusion model to market data and price our barrier options. The aim of this thesis is to decribe and to compare different methods of valuating barrier options. 1

Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications Master Thesis - Petr Čoupek Abstract In this thesis, we introduce a stochastic integral of deterministic Hilbert space valued functions driven by a Gaussian process of the Volterra form βt = t 0 K(t, s)dWs, where W is a Brownian motion and K is a square integrable kernel. Such processes generalize the fractional Brownian motion BH of Hurst parameter H ∈ (0, 1). Two sets of conditions on the kernel K are introduced, the singular case and the regular case, and, in particular, the regular case is studied. The main result is that the space H of β-integrable functions can be, in the strictly regular case, embedded in L 2 1+2α ([0, T]; V ) which corresponds to the space L 1 H ([0, T]) for the fractional Brownian mo- tion. Further, the cylindrical Gaussian Volterra process is introduced and a stochastic integral of deterministic operator-valued functions, driven by this process, is defined. These results are used in the theory of stochastic differential equations (SDE), in particular, measurability of a mild solution of a given SDE is proven.

Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.

Cette thèse comporte deux chapitres principaux. Deux problèmes indépendants de Modélisation Mathématique y sont étudiés. Au chapitre 1, on étudiera le problème de l’existence et de l’unicité des distributions quasi-stationnaires (DQS) pour un mouvement Brownien avec dérive, tué en zéro dans le cas où la frontière d’entrée est l’infini et la frontière de sortie est zéro selon la classification de Feller.Ce travail est lié à l’article pionnier dans ce sujet  par Cattiaux, Collet, Lambert, Martínez, Méléard, San Martín; où certaines conditions suffisantes ont été établies pour prouver l’existence et l’unicité de DQS dans le contexte d’une famille de Modèles de Dynamique des Populations.Dans ce chapitre, nous généralisons les théorèmes les plus importants de ce travail pionnier, la partie technique est basée dans la théorie de Sturm-Liouville sur la demi-droite positive. Au chapitre 2, on étudiera le problème d’obtenir des bornes inférieures optimales sur l’Hamiltonien du Modèle d’Ising avec interactions à longue portée, l’interaction entre deux spins situés à distance d décroissant comme d^(2-a), où a ϵ[0,1).Ce travail est lié à l’article publié en 2005 par Cassandro, Ferrari, Merola, Presutti où les bornes inférieures optimales sont obtenues dans le cas où a est dans [0,(log3/log2)-1) en termes de structures hiérarchiques appelées triangles et contours.Les principaux théorèmes obtenus dans cette thèse peuvent être résumés de la façon suivante:1. Il n’existe pas de borne inférieure optimale pour l’Hamiltonien en termes de triangles pour a dans ϵ[log2/log3,1). 2. Il existe une borne optimale pour l’Hamiltonien en termes de contours pour a dans a ϵ [0,1). / This thesis contains two main Chapters, where we study two independent problems of Mathematical Modelling : In Chapter 1, we study the existence and uniqueness of Quasi Stationary Distributions (QSD) for a drifted Browian Motion killed at zero, when $+infty$ is an entrance Boundary and zero is an exit Boundary according to Feller's classification. The work is related to the previous paper published in 2009 by { Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., San Martín, where some sufficient conditions were provided to prove the existence and uniqueness of QSD in the context of a family of Population Dynamic Models. This work generalizes the most important theorems of this work, since no extra conditions are imposed to get the existence, uniqueness of QSD and the existence of a Yaglom limit. The technical part is based on the Sturm Liouville theory on the half line. In Chapter 2, we study the problem of getting quasi additive bounds on the Hamiltonian for the Long Range Ising Model when the interaction term decays according to d^{2-a}, a ϵ[0,1). This work is based on the previous paper written by Cassandro, Ferrari, Merola, Presutti, where quasi-additive bounds for the Hamiltonian were obtained for a in [0,(log3/log2)-1) in terms of hierarchical structures called triangles and Contours. The main theorems of this work can be summarized as follows: 1 There does not exist a quasi additive bound for the Hamiltonian in terms of triangles when a ϵ [0,(log3/log2)-1), 2. There exists a quasi additive bound for the Hamiltonian in terms of Contours for a in [0,1).