In this masterthesis I have rewied basic approaches to volatility estimating. These approaches are based on classical and Bayesian statistics. I have applied the volatility models for the purpose of volatility forecasting of a different foreign exchange (EURUSD, GBPUSD and CZKEUR) in the different period (from a second period to a day period). I formulate the models EWMA, GARCH, EGARCH, IGARCH, GJRGARCH, jump diffuison with constant volatility and jump diffusion model with stochastic volatility. I also proposed an MCMC algorithm in order to estimate the Bayesian models. All the models we estimated as univariate models. I compared the models according to Mincer Zarnowitz regression. The most successfull model is the jump diffusion model with a stochastic volatility. On the second place they were the GJR- GARCH model and the jump diffusion model with a constant volatility. But the jump diffusion model with a constat volatilit provided much more overvalued results.The rest of the models were even worse. From the rest the IGARCH model is the best but provided undervalued results. All these findings correspond with R squared coefficient.

L’homme est un animal social. La majorité des décisions que nous prenons se font dans un contexte social et dépendent d’autrui, ce qui implique des calculs cérébraux complexes qui incluent tous les facteurs contextuels et environnementaux. La majorité des études ultérieures de la prise en compte d’autrui dans la décision ont utilisé des tâches de partage de récompenses entre soi et autrui. Les choix possibles amènent le décideur à considérer autrui, mais dans le but de gagner soi-même une récompense ; donc dans un contexte où les récompenses liées à soi et les récompenses liées à autrui sont confondues. Le travail présenté dans cette thèse avait pour but une meilleure compréhension des mécanismes cérébraux soutenant l’intégration d’autrui dans la prise de décision, sans que la récompense pour autrui n’interfère directement avec soi. Nous nous sommes appuyés sur le cadre théorique de la décision perceptuelle et des modèles de diffusion pour l'étude i) des modifications du processus décisionnel induites par une récompense monétaire allant à autrui et ii) de l’impact de l’effet d’audience (le fait de se sentir observé) sur la décision. Nos résultats computationnels montrent qu'une récompense pour autrui, par rapport à une récompense pour soi, et une audience, par rapport au secret, modifient le taux de dérive de la variable de décision. En magnétoencéphalographie, nos résultats indiquent que les décisions pour soi et pour autrui diffèrent pendant, mais aussi après, la prise de décision dans des zones cérébrales associées avec la transformation sensori-motrice, l'ajustement du compromis entre rapidité et justesse et avec la cognition sociale. Ainsi, le cortex temporal montre des différences de -1170 millisecondes (ms) à -1023 ms, de -993 ms à -915 ms et de -343 ms à -188 ms en amont de la réponse. Ce qui suppose une influence sur l’intégration des preuves sensorielles. Après la décision, les régions frontales ont également montré des différences entre soi et autrui, de 153 ms à 303 ms post-réponse, suggérant une différence entre soi et autrui dans l’ajustement du compris entre justesse et rapidité. Le bénéficiaire de la récompense associée à la décision modifie les paramètres décisionnels et les corrélats cérébraux de la décision perceptuelle, démontrant l’importance du contexte social dans l’implémentation de la prise de décision chez l’Homme. Ce travail appuie également l’utilité des modèles mathématiques tels que les modèles de diffusion dans la compréhension des processus décisionnels, même de ceux découlant de la cognition sociale / Humans are inherently social: most of human’s decisions are within a social context and depend on others. For more than a century, researchers explore aspects of social cognition. Aiming to understand human behavior in social contexts, neuro-economic researches showed that taking others into account involve complex brain computations that include all environmental and contextual factors. However, most of the work was made using money allocation tasks; mixing self-affecting and other-affecting rewards into the decision making process. The present work intended the understanding of the brain mechanisms underpinning the integration of others into the decision making process for decisions that include others and do not interfere with self-rewards.Taking advantage of mathematical models from the drift diffusion models framework, we conducted experiments investigating how others influence the mechanistic of perceptual decisions and their correlates in the human brain. We showed that taking rewards for others into account and being observed by others influence the drift rate of the decision variable. The drift rate is higher in audience than in secret and higher for self-rewards than for other-rewards. These results indicate that others are integrated into the accumulation process together with the evidence available for making a decision. At the brain level, we found difference between self and other decisions over the anterior temporal and centro-frontal cortices during decision making. This suggests that the beneficiary of a decision modifies sensory-motor transformation processes. In addition, self- and other-affecting difference showed difference over the medial frontal sensors after the decision making process, indicating a variation in the speed-accuracy tradeoff adjustment process

>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.

Magnetic resonance (MR) images together with computed tomography (CT) images are used in many medical practices, such as radiation therapy. To capture those images, patients have to undergo two separate scans: one for the MR image, which involves using strong magnetic fields, and one for the CT image which involves using radiation (x-rays). Another approach is to generate synthetic CT (sCT) images from MR images, thus the patients only have to take one image (the MR image), making the whole process easier and more effcient. One way of generating sCT images is by using generative diffusion models which are a relatively new class in generative models. To this end, this project aims to enquire whether generative diffusion models are capable of generating viable and realistic sCT images from MR images. Firstly, a denoising diffusion probabilistic model (DDPM) with a U-Net backbone neural network is implemented and tested on the MNIST dataset, then it is implemented on a pelvis dataset consisting of 41600 pairs of images, where each pair is made up of an MR image with its respective CT image. The MR images were added at each sampling step in order to condition the sampled sCT images on the MR images. After successful implementation and training, the developed diffusion model got a Fréchet inception distance (FID) score of 14.45, and performed as good as the current state-of-the-art model without any major optimizations to the hyperparameters or to the model itself. The results are very promising and demonstrate the capabilities of this new generative modelling framework.

Объект исследования: процесс аугментации изображений для решения задачи сегментации. Предмет исследования: методы аугментации и машинного обучения, с помощью которых осуществляется сегментация изображений. Цель работы: исследование эффективности генеративной аугментации изображений, выполненной с помощью диффузионной модели Stable Diffusion на примере задачи семантической сегментации. В процессе исследования проводились: рассмотрение основных подходов сегментации изображений и методов аугментации данных, разработка и реализация экспериментов для оценки эффективности генеративной аугментации изображений. В работе продемонстрирована эффективность подхода аугментации изображений, реализованного за счет расширения части исходного датасета путем генерирования новых данных с помощью диффузионной модели. Область практического применения: предложенный подход может быть использован для улучшения качества работы моделей семантической сегментации изображений в условиях ограниченного количества исходных данных, дефицита размеченных данных или дисбаланса данных. / Object of study: the process of image augmentation to solve the segmentation problem. Subject of research: augmentation and machine learning methods used for image segmentation. Purpose of the work: to study the effectiveness of generative image augmentation performed using the Stable Diffusion model using the example of a semantic segmentation task. During the research process, the following was carried out: consideration of the main approaches to image segmentation and data augmentation methods, development and implementation of experiments to evaluate the effectiveness of generative image augmentation. The work demonstrates the effectiveness of the image augmentation approach, implemented by expanding part of the original dataset by generating new data using a diffusion model. Area of practical application: the proposed approach can be used to improve the quality of work of semantic image segmentation models in conditions of a limited amount of source data, a shortage of labeled data, or data imbalance.

Wir studieren das Verhalten des Maximums, des Minimums und des Endwerts zeithomogener eindimensionaler Diffusionen auf endlichen Zeitintervallen. Zuerst beweisen wir mit Hilfe des Malliavin-Kalküls ein Existenzresultat für die gemeinsamen Dichten. Außerdem leiten wir Entwicklungen der gemeinsamen Momente des Tripels (H,L,X) zur Zeit Delta bzgl. Delta her. Dabei steht X für die zugrundeliegende Diffusion, und H und L bezeichnen ihr fortlaufendes Maximum bzw. Minimum. Ein erster Ansatz, der vollständig auf den elementaren Abschätzungen der Doob’schen und der Cauchy-Schwarz’schen Ungleichung beruht, liefert eine Entwicklung bis zur Ordnung 2 bzgl. der Wurzel der Zeitvariablen Delta. Ein komplexerer Ansatz benutzt Partielle-Differentialgleichungstechniken, um eine Entwicklung der einseitigen Austrittswahrscheinlichkeit für gepinnte Diffusionen zu bestimmen. Da eine Entwicklung der Übergangsdichten von Diffusionen bekannt ist, erhält man eine vollständige Entwicklung der gemeinsamen Wahrscheinlichkeit von (H,X) bzgl. Delta. Die entwickelten Verteilungseigenschaften erlauben es uns, eine Theorie für Martingalschätzfunktionen, die aus wertebereich-basierten Daten konstruiert werden, in einem parameterisierten Diffusionsmodell, herzuleiten. Ein Small-Delta-Optimalitätsansatz, der die approximierten Momente benutzt, liefert eine Vereinfachung der vergleichsweise komplizierten Schätzprozedur und wir erhalten asymptotische Optimalitätsresultate für gegen 0 gehende Sampling-Frequenz. Beim Schätzen des Drift-Koeffizienten ist der wertebereich-basierte Ansatz der Methode, die auf equidistanten Beobachtungen der Diffusion beruht, nicht überlegen. Der Effizienzgewinn im Fall des Schätzens des Diffusionskoeffizienten ist hingegen enorm. Die Maxima und Minima in die Analyse miteinzubeziehen senkt die Varianz des Schätzers für den Parameter in diesem Szenario erheblich. / We study the behavior of the maximum, the minimum and the terminal value of time-homogeneous one-dimensional diffusions on finite time intervals. To begin with, we prove an existence result for the joint density by means of Malliavin calculus. Moreover, we derive expansions for the joint moments of the triplet (H,L,X) at time Delta w.r.t. Delta. Here, X stands for the underlying diffusion whereas H and L denote its running maximum and its running minimum, respectively. In a first approach that entirely relies on elementary estimates, such as Doob’s inequality and Cauchy-Schwarz’ inequality, we derive an expansion w.r.t. the square root of the time parameter Delta including powers of 2. A more sophisticated ansatz uses partial differential equation techniques to determine an expansion of the one-barrier hitting time probability for pinned diffusions. For an expansion of the transition density of diffusions is known, one obtains an overall expansion of the joint probability of (H,X) w.r.t. Delta. The developed distributional properties enable us to establish a theory for martingale estimating functions constructed from range-based data in a parameterized diffusion model. A small-Delta-optimality approach, that uses the approximated moments, yields a simplification of the relatively complicated estimating procedure and we obtain asymptotic optimality results when the sampling frequency Delta tends to 0. When it comes to estimating the drift coefficient the range-based method is not superior to the method relying on equidistant observations of the underlying diffusion alone. However, there is an enormous gain in efficiency at the estimation for the diffusion coefficient. Incorporating the maximum and the minimum into the analysis significantly lowers the asymptotic variance of the estimators for the parameter in this scenario.

Die vorliegende Arbeit befasst sich mit Innovationsdiffusionsmodellen und deren Anwendung in der Marketingpraxis. Sie hat zwei Ziele: Einen Überblick über existierende Innovationsmodelle zu schaffen und ein neues besseres Modell zu entwickeln. Es wird ein neuer Klassifizierungsansatz vorgeschlagen, mit dessen Hilfe ein strukturierter Überblick über die vorhandenen zahlreichen Innovationsdiffusionsmodelle möglich wird. Die Klassifizierung beruht auf den Annahmen in den Innovationsdiffusionsmodellen. Dies erlaubt im Gegensatz zu den bekannten Klassifizierungen (z.B. von Roberts/Lattin (2000)) die Bildung von disjunkten Modellklassen. Anhand der neuen Klassifizierung werden die prominenten Modelle, wie z.B. Bass-Modell (1969) bzw. Kalish-Modell (1985) eingeordnet und ihre Vor- und Nachteile aufgezeigt. Dieser Ansatz erleichtert eine Entscheidung für das beste zu verwendende Modell, wenn bekannt ist, welche Daten (Absatzdaten, Daten über Konsumenten etc.) zur Verfügung stehen und/oder welches Ziel (Absatzprognose, Preisbestimmung) verfolgt wird. Im zweiten Teil der Arbeit wird ein neues hybrides Innovationsdiffusionsmodell – das Information-Disicion-Evaluation-Modell (IED-Modell) – vorgestellt. Das IED-Modell besitzt zahlreiche Vorteile gegenüber existierenden Innovationsdiffusionsmodellen. Die Struktur des IED-Modells ist sehr allgemein so, dass das IED-Modell als eine Modellklasse bezeichnet werden könnte. Werden die Annahmen des IED-Modells genau definiert (z.B. über die Anzahl der Wettbewerbsprodukte usw.), erhält es eine explizite Form, die prominenten Innovationsdiffusionsmodellen ähnlich oder vollkommen identisch sein kann (für die Erstellung einer expliziten Form des IED-Modells siehe www.ied-modell.de). Ein solcher allgemeiner Modellierungsansatz des IED-Modells ist neu für die Innovationsdiffusionsforschung. Das IED-Modell und dessen Annahmen werden mittels Monte-Carlo-Simulationen analysiert. Beim empirischen Test an realen Daten wird das IED-Modell mit vier renommierten Innovationsdiffusionsmodellen verglichen. Laut diesem Vergleich ist das Anpassungsvermögen des IED-Modells im Durchschnitt besser als das der vier Vergleichsmodelle. Bei drei- und zehnmonatlichen Prognosen zeigte das IED-Modell eine sehr gute Vorhersagefähigkeit. / This work assesses innovation diffusion models and their application in marketing management. Its two principal aims are: (1) to give an overview of existing innovation diffusion models and (2) to develop a new and improved model. A new classification approach is proposed. The classification methodology bases on typical assumptions made in innovation diffusion models. Unlike prior classifications, e.g. Roberts/Lattin (2000), this approach allows for disjunctive classes. By means of this classification renowned models like Bass Model (1969) or Kalish Model (1985) are categorized, and their advantages and disadvantages are analyzed. This helps decide which model should be used depending on data availability (sales data, consumer data etc.) and the overall goal of a model investigation (sales forecast, pricing etc.). In the second part of this work the new hybrid innovation diffusion model – Innovation-Decision-Evaluation model (IED model) – is described. The model has several advantages compared with existing models. The structure of the IED Model is non-specific so that the IED model can be described as a distinct model class. When assumptions of the IED model are specified (e.g. number of competitive products) the model gets an explicit form which can be similar or even identical to other innovation diffusion models (for the design of an explicit model form see also www.ied-modell.de). Such a generalized modeling approach in IED modelling is new in innovation diffusion research. The IED model and its assumptions are analysed with Monte Carlo simulations. Its results are also empirically tested and compared with four renowned innovation diffusion models. The comparison reveals that the IED model has the best average fit and good forecast goodness.

Durant cette thèse, nous nous sommes intéressés aux outils permettant d'extraire les propriétés structurelles et temporelles de réseaux dynamiques ainsi que les caractéristiques de certains scénarios de diffusion pouvant s'opérer sur ces réseaux. Nous avons travaillé sur un jeu de données spécifiques, issu du projet MOSAR, qui comporte entre autre le réseau de proximité des personnes au cours du temps durant 6 mois à l'hôpital de Berk-sur-mer. Ce réseau est particulier dans le sens où il est constitué de trois dimensions: temporelle, structurelle par la répartition des personnes en services et fonctionnelle car chaque personne appartient à une catégorie socio-professionnelle. Pour chacune des dimensions, nous avons utilisé des outils existants en physique statistique ainsi qu'en théorie des graphes pour extraire des informations permettant de décrire certaines propriétés du réseau. Cela nous a permis de souligner le caractère très structuré de la répartition des contacts qui suit la répartition en services et mis en évidence les accointances entre certaines catégories professionnelles. Concernant la partie temporelle, nous avons mis en avant l'évolution périodique circadienne et hebdomadaire ainsi que les différences fondamentales entre l'évolution des interactions des patients et celle des personnels. Nous avons aussi présenté des outils permettant de comparer l'activité entre deux périodes données et de quantifier la similarité de ces périodes. Nous avons ensuite utilisé la technique de simulation pour extraire des propriétés de diffusion de ce réseau afin de donner quelques indices pour établir une politique de prévention. / In this thesis, we focus on tools whose aim is to extract structural and temporal properties of dynamic networks as well as diffusion characteristics which can occur on these networks. We work on specific data, from the European MOSAR project, including the network of individuals proximity from time to time during 6 months at the Brek-sur-Mer Hospital. The studied network is notable because of its three dimensions constitution : the structural one induced by the distribution of individuals into distinct services, the functional dimension due to the partition of individual into groups of socio-professional categories and the temporal dimension.For each dimension, we used tools well known from the areas of statistical physics as well as graphs theory in order to extract information which enable to describe the network properties. These methods underline the specific structure of the contacts distribution which follows the individuals distribution into services. We also highlight strong links within specific socio-professional categories. Regarding the temporal part, we extract circadian and weekly patterns and quantify the similarities of these activities. We also notice distinct behaviour within patients and staff evolution. In addition, we present tools to compare the network activity within two given periods. To finish, we use simulations techniques to extract diffusion properties of the network to find some clues in order to establish a prevention policy.

In this thesis, we consider stochastic systems which arise in different real-world application contexts. The first problem we consider is based on product adoption and pricing. A monopolist selling a product has to appropriately price the product over time in order to maximize the aggregated profit. The demand for a product is uncertain and is influenced by a number of factors, some of which are price, advertising, and product technology. We study the influence of price on the demand of a product and also how demand affects future prices. Our approach involves mathematically modelling the variation in demand as a function of price and current sales. We present a simulation-based algorithm for computing the optimal price path of a product for a given period of time. The algorithm we propose uses a smoothed-functional based performance gradient descent method to find a price sequence which maximizes the total profit over a planning horizon. The second system we consider is in the domain of sensor networks. A sensor network is a collection of autonomous nodes, each of which senses the environment. Sensor nodes use energy for sensing and communication related tasks. We consider the problem of finding optimal energy sharing policies that maximize the network performance of a system comprising of multiple sensor nodes and a single energy harvesting(EH) source. Nodes periodically sense a random field and generate data, which is stored in their respective data queues. The EH source harnesses energy from ambient energy sources and the generated energy is stored in a buffer. The nodes require energy for transmission of data and and they receive the energy for this purpose from the EH source. There is a need for efficiently sharing the stored energy in the EH source among the nodes in the system, in order to minimize average delay of data transmission over the long run. We formulate this problem in the framework of average cost infinite-horizon Markov Decision Processes[3],[7]and provide algorithms for the same.

In the area of Traffic Sign Recognition (TSR), deep learning models are trained to detect and classify images of traffic signs. The amount of data available to train these models is often limited, and collecting more data is time-consuming and expensive. A possible complement to traditional data acquisition, is to generate synthetic images with a generative machine learning model. This thesis investigates the use of denoising diffusion probabilistic models for generating synthetic data of one or multiple traffic sign classes, when providing different amount of real images for that class (classes). In the few-sample method, the number of images used was from 1 to 1000, and zero images were used in the zero-shot method. The results from the few-sample method show that combining synthetic images with real images when training a traffic sign classifier, increases the performance in 3 out of 6 investigated cases. The results indicate that the developed zero-shot method is useful if further refined, and potentially could enable generation of realistic images of signs not seen in the training data.