<p>We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and S{\o}rensen (2007), another one, from  Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points!</p>

Cette thèse porte sur l’analyse statistique de quelques modèles de processus stochastiques gouvernés par des bruits de type fractionnaire, en temps discret ou continu.Dans le Chapitre 1, nous étudions le problème d’estimation par maximum de vraisemblance (EMV) des paramètres d’un processus autorégressif d’ordre p (AR(p)) dirigé par un bruit gaussien stationnaire, qui peut être à longue mémoire commele bruit gaussien fractionnaire. Nous donnons une formule explicite pour l’EMV et nous analysons ses propriétés asymptotiques. En fait, dans notre modèle la fonction de covariance du bruit est supposée connue, mais le comportement asymptotique de l’estimateur (vitesse de convergence, information de Fisher) n’en dépend pas.Le Chapitre 2 est consacré à la détermination de l’entrée optimale (d’un point de vue asymptotique) pour l’estimation du paramètre de dérive dans un processus d’Ornstein-Uhlenbeck fractionnaire partiellement observé mais contrôlé. Nous exposons un principe de séparation qui nous permet d’atteindre cet objectif. Les propriétés asymptotiques de l’EMV sont démontrées en utilisant le programme d’Ibragimov-Khasminskii et le calcul de transformées de Laplace d’une fonctionnellequadratique du processus.Dans le Chapitre 3, nous présentons une nouvelle approche pour étudier les propriétés du mouvement brownien fractionnaire mélangé et de modèles connexes, basée sur la théorie du filtrage des processus gaussiens. Les résultats mettent en lumière la structure de semimartingale et mènent à un certain nombre de propriétés d’absolue continuité utiles. Nous établissons l’équivalence des mesures induites par le mouvement brownien fractionnaire mélangé avec une dérive stochastique, et en déduisons l’expression correspondante de la dérivée de Radon-Nikodym. Pour un indice de Hurst H > 3=4, nous obtenons une représentation du mouvement brownien fractionnaire mélangé comme processus de type diffusion dans sa filtration naturelle et en déduisons une formule de la dérivée de Radon-Nikodym par rapport à la mesurede Wiener. Pour H < 1=4, nous montrons l’équivalence de la mesure avec celle la composante fractionnaire et obtenons une formule pour la densité correspondante. Un domaine d’application potentielle est l’analyse statistique des modèles gouvernés par des bruits fractionnaires mélangés. A titre d’exemple, nous considérons le modèle de régression linéaire de base et montrons comment définir l’EMV et étudié son comportement asymptotique. / This thesis focuses on the statistical analysis of some models of stochastic processes generated by fractional noise in discrete or continuous time.In Chapter 1, we study the problem of parameter estimation by maximum likelihood (MLE) for an autoregressive process of order p (AR (p)) generated by a stationary Gaussian noise, which can have long memory as the fractional Gaussiannoise. We exhibit an explicit formula for the MLE and we analyze its asymptotic properties. Actually in our model the covariance function of the noise is assumed to be known but the asymptotic behavior of the estimator ( rate of convergence, Fisher information) does not depend on it.Chapter 2 is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. We expose a separation principle that allows us toreach this goal. Large sample asymptotical properties of the MLE are deduced using the Ibragimov-Khasminskii program and Laplace transform computations for quadratic functionals of the process.In Chapter 3, we present a new approach to study the properties of mixed fractional Brownian motion (fBm) and related models, based on the filtering theory of Gaussian processes. The results shed light on the semimartingale structure andproperties lead to a number of useful absolute continuity relations. We establish equivalence of the measures, induced by the mixed fBm with stochastic drifts, and derive the corresponding expression for the Radon-Nikodym derivative. For theHurst index H > 3=4 we obtain a representation of the mixed fBm as a diffusion type process in its own filtration and derive a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H < 1=4, we prove equivalenceto the fractional component and obtain a formula for the corresponding derivative. An area of potential applications is statistical analysis of models, driven by mixed fractional noises. As an example we consider only the basic linear regression setting and show how the MLE can be defined and studied in the large sample asymptotic regime.

Ce travail de thèse porte sur la modélisation et la simulation numérique de la collision et l'agglomération de particules colloïdales dans un écoulement fluide turbulent par une nouvelle méthode. Ces particules sont sensibles dans une même mesure aux effets brownien et turbulent. La première partie du travail concerne la modélisation du phénomène physique,allant du transport des particules jusqu'à la modélisation des forces d'adhésion physico-chimiques en passant par l'étape cruciale qui est la détection des interactions entre les particules (collisions). Cette détection des collisions est dans un premier temps étudiée par rapport aux algorithmes classiques existants dans la littérature. Bien que très efficaces dans le cadre de particules soumises à l'agitation turbulente, les conclusions de cette partie exposent les limites des méthodes existantes en termes de coûts numériques, pour le traitement d'un ensemble de colloïdes soumis au mouvement brownien. La seconde partie du travail oriente alors les travaux vers une vision novatrice du phénomène physique considéré. Le caractère diffusif aléatoire est alors considéré d'un point de vu stochastique, comme un processus conditionné dans l'espace et dans le temps. Ainsi, une nouvelle méthode de détection et de traitement des collisions de particules soumises exclusivement à un mouvement diffusif est présentée et validée, exposant un gain considérable en termes de coûts numériques. Le potentiel de cette nouvelle approche est validé et ouvre de nombreuses pistes de réflexion dans l'utilisation des méthodes stochastiques appliqués à la représentation de la physique / Ph.D thesis focuses on modeling and numerical simulation of collision and agglomeration of colloidal particles in a turbulent flow by using a new method. These particles are affected by both Brownian and turbulent effects. The first part of the work deals with current models of the physical phenomenon, from the transport of single particles to a model for physico-chemical adhesive forces, and points out the critical step which is the detection of interactions between particles (collisions). This detection is initially studied by applying classical algorithms existing in the literature. Although they are very efficient in the context of particles subject to turbulent agitation, first conclusions show the limitations of these existing methods in terms of numerical costs, considering the treatment of colloids subject to the Brownian motion. The second part of this work proposes a new vision of the physical phenomenon focusing on the random diffusive behaviour. This issue is adressed from a stochastic point of view as a process conditionned in space and time. Thus, a new method for the detection and treatment of collisions is presented and validated, which represents considerable gain in terms of numerical cost. The potential of this new approach is validated and opens new opportunities for the use of stochastic methods applied to the representation of physics

>Magister Scientiae - MSc / Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

The paper examines the impact of fees on the return of a fund investment using different simulation and fee structure models. The results show that fees have a significant expected impact, particularly for well-performing funds. Two simulation models were used, the Geometric Brownian Motion (GBM) model and Merton Jump Diffusion (MJD) model. Two fee structures were also analysed for each simulation, a High-water mark fee structure and a Hurdle fee structure. Comparing the GBM and MJD models, the two tend to generate very similar fee statistics even though the MJD model's day-to-day returns fit better with empirical data. When comparing the HWM and Hurdle fee models, larger differences are observed. While overall average fee statistics are similar, the performance fee statistics are significantly higher in the Hurdle fee structure for assets achieving higher returns, e.g. at least an 8% annual return. However, the HWM fee structure tends to generate higher performance fees for assets with low returns. Regression models are also developed for each combination of the simulation model and fee structure. The regression models reflect the above conclusions and can for investors serve as simple key indicators to estimate expected fund fee payments. The GBM regression results are likely more useful than the MJD regression results, as the parameters of the former are easier to calculate based on historical return data. / Uppsatsen undersöker effekten av avgifter på avkastningen av en fondinvestering med hjälp av olika simuleringar och avgiftsmodeller. Resultaten visar att avgifter förväntas ha en betydande påverkan, särskilt för fonder som genererar hög avkastning. Två simuleringar användes, Geometric Brownian Motion (GBM) och Merton Jump Diffusion (MJD). Två avgiftsstrukturer analyserades också för varje simulering, en High-water mark avgiftsstruktur och en Hurdle avgiftsstruktur. Jämförelse mellan GBM och MJD-modellerna visar att de två tenderar att generera mycket liknande avgiftsstatistik trots att MJD-modellens dagliga avkastning passar bättre med empiriska data. Vid jämförelse av HWM- och Hurdle avgiftsmodellerna observeras större skillnader. Medan den övergripande genomsnittliga avgiftsstatistiken är liknande för avgiftsmodellerna, är resultatbaserade avgifterna betydligt högre i Hurdle avgiftsstrukturen för tillgångar som uppnår högre avkastning, t.ex. minst 8% årlig avkastning. Däremot tenderar HWM-avgiftsstrukturen att generera högre resultatbaserade avgifter för tillgångar med låg avkastning. Regressionsmodeller utvecklades också för varje kombination av simulering och avgiftsstruktur. Regressionmodellerna återspeglar ovanstående slutsatser och kan för investerare fungera som enkla nyckeltal för att uppskatta förväntad kostnad av fondavgifter. GBM-regressionsresultaten är sannolikt mer användbara än MJD-regressionsresultaten, eftersom parametrarna för den förra är lättare att beräkna baserat på historisk avkastningsdata.

The fundamental issue of determining acoustic properties of porous media from their local geometry is examined in this PhD dissertation thesis, thanks to a sample of open-cell aluminum foam analyzed by axial computed microtomography. Various geometric properties are measured to characterize the experimental sample at the cell size level. This is done in order to reconstruct a porous medium by means of idealized three- and two- dimensional unit-cells.The frequency dependant thermal and velocity fields governing the propagation and dissipation of acoustic waves through rigid porous media are computed by Brownian motion simulation and the finite element method, respectively. Macroscopic behavior is derived by spatial averaging of the local fields. Our results are compared to experimental data obtained from impedance tube measurements. Firstly, this approach leads to the identification of the macroscopic parameters involved in Pride and Lafarge semiphenomenological models. Secondly, it yields a direct access to thermal and viscous dynamic permeabilities. However, the bi-dimensional model underestimates the static viscous permeability as well as the viscous characteristic length; what thus require a three-dimensional implementation.

The thermal velocity fluctuations of microscopic particles mediate the transition from microscopic statistical mechanics to macroscopic long-time diffusion. Prior to this work, detection methods lacked the sensitivity necessary to resolve motion at the length and time scales at which thermal velocity fluctuations occur. This dissertation details two experiments which resulted in velocity measurement of the thermal motion of dielectric microspheres suspended by an optical trap in gases and liquids. First, optical tweezers were used to trap glass microspheres in air over a wide range of pressures and a detection system was developed to track the trapped microspheres' trajectories with MHz bandwidth and <100 fm/rt(Hz) position sensitivity. Low-noise trajectory measurements allowed for observation of fluctuations in the instantaneous velocity of a trapped particle with a signal to noise ratio (SNR) of 26 dB, and provided direct verification of the equipartition theorem and of the Maxwell-Boltzmann velocity distribution for a single Brownian particle. Next, the detection technology was further optimized and used to track optically trapped silica and barium titanate glass microspheres in water and acetone with >50 MHz bandwidth and <3 fm/rt(Hz) sensitivity. Brownian motion in a liquid is influenced by hydrodynamic, time-retarded coupling between the particle and the fluid flow its motion generates. Our measurements allowed for instantaneous velocity measurement with an SNR of up to 16 dB and confirmed the Maxwell Boltzmann distribution for Brownian motion in a liquid. The measurements also revealed several unusual features predicted for Brownian motion in the regime of hydrodynamic coupling, including faster-than-exponential decay of the velocity autocorrelation function, correlation of the thermal force and non-zero cross-correlation between the particle's velocity and the thermal force preceding it. / text

We use simultaneous observation of translational and rotational Brownian motion of domains in lipid membranes to test the hydrodynamics-based theory for the viscous drag on the membrane inclusion. We find that translational and rotational diffusion coefficients of micrometer-sized solid (gel-phase) domains in giant unilamellar vesicles showing fluid–gel phase coexistence are in excellent agreement with the theoretical predictions. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

The main contribution of the present thesis is in two aspects. The first one, which is the heart of the thesis, is to explore the fundamental relation between rough paths and their signatures. Our main goal is to give a geometric characterization of the kernel of the signature map in different situations. In Chapter Two, we start by establishing a general fact that a continuous Jordan curve on a Riemannian manifold can be arbitrarily well approximated by piecewise minimizing geodesic interpolations which are again Jordan. This result enables us to prove a generalized version of Green’s theorem for planar Jordan curves with finite p-variation 1 &le; p < 2, and to prove that two such Jordan curves have the same signature if and only if they are equal up to reparametrization. In Chapter Three, we investigate the problem for general weakly geometric rough paths. In particular, we show that a weakly geometric rough path has trivial signature if and only if it is tree-like in the sense we will define later on. In Chapter Four, we study the problem in the probabilistic setting. In particular, we show that for a class of stochastic processes, with probability one the sample paths are determined by their signatures up to reparametrization. A fundamental example is Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge. The second one is an application of rough path theory to the study of nonlinear diffusions on manifolds under the framework of nonlinear expectations. In Chapter Five, we begin by studying the geometric rough path nature of G-Brownian motion. This enables us to introduce rough differential equations driven by G-Brownian motion from a pathwise point of view. Next we establish the fundamental relation between rough (pathwise theory) and stochastic (L²-theory) differential equations driven by G-Brownian motion. This is a crucial point of understanding nonlinear diffusions and their generating heat flows on manifolds from an intrinsic point of view. Finally, from the pathwise point of view we construct G-Brownian motion on a compact Riemannian manifold and establish its generating heat flow for a class of G-functions under orthogonal invariance. As an independent interest, we also develop the Euler-Maruyama scheme for stochastic differential equations driven by G-Brownian motion.

Le travail réalisé dans cette thèse porte sur l’étude du mouvement Brownien d’une suspension colloïdale sous champ de force optique faible et l’étude fondamentale des effets convectifs et de diffusion anormale. Nous avons construit un microscope à fond noir afin de suivre les particules et de reconstruire leurs trajectoires avec une résolution spatiale de 20 nm et une résolution temporelle de 8 ms. Ces trajectoires sont analysées statistiquement afin d’en extraire la contribution balistique induite par la force de pression de radiation appliquée par le laser d’illumination. En plus de l’effet mécanique du laser sur les particules, le fluide absorbe les radiations ce qui le chauffe et crée ainsi une différence de température entre la partie illuminée et la partie non illuminée de l’échantillon.Nous validons aussi les hypothèses de stationnarité et d’érgodicité qui sont fondamentales pour notre stratégie de mesure de force faible. L’analyse statistique fine de notre système nous permet de mettre en évidence et de caractériser des effets de diffusion anormale brownienne. Nos expériences révèlent en effet la présence de trajectoires anormales dont l’origine se comprend comme un effet d’interaction entre la particule suivie et le reste de l’ensemble colloïdal. / The work presented in this thesis deals with the study of the Brownian motion of a colloidal suspension under an external weak optical force, the study of convective effects and anomalous diffusion. We have built a dark field microscope in order to track the particles and reconstruct the Brownian trajectories with a spatial resolution of 20 nm and a temporal resolution of 8 ms.Statistical analysis of the trajectories has allowed us to extract the ballistic contribution induced by the radiation pressure force exerted by irradiating a laser on the particles. In addition to the mechanical effect of the laser on the particles, the fluid absorbs the radiation. Consequently, the temperature of the fluid rises and results in a thermal difference between the illuminated and the non-illuminated areas of the sample. In order to validate our weak force measurement, we have investigated two fundamental hypotheses in statistical physics: ergodicity and stationary aspect. A closer statistical analysis enables us to demonstrate and characterize the effect of anomalous Brownian diffusion. Our experiments have revealed the existence of anomalous trajectories, which can be understood as an effect of the interactions between the particles.