In this paper, we discuss methods for developing a stochastic model which incorporates behavior differences in the predation movements of Anelosimus studiosus (a subsocial spider). Stochastic models for animal movement and, in particular, spider predation movement have been developed previously; however, this paper focuses on the development and implementation of the necessary mathematical and statistical methods required to expand such a model in order to capture a variety of distinct behaviors. A least squares optimization algorithm is used for parameter estimation to fit a single stochastic model to an individual spider during predation resulting in unique parameter values for each spider. Similarities and variations between parameter values across the spiders are analyzed and used to estimate probability distributions for the variable parameter values. An aggregate stochastic model is then created which incorporates the individual dynamics. The comparison between the optimal individual models to the aggregate model indicate the methodology and algorithm developed in this paper are appropriate for simulating a range of individualistic behaviors.

In this paper, we develop a stochastic differential equation model to simulate the movement of a social/subsocial spider species, Anelosimus studiosus, during prey capture using experimental data collected in a structured environment. In a subsocial species, females and their maturing offspring share a web and cooperate in web maintenance and prey capture. Furthermore, observations indicate these colonies change their positioning throughout the day, clustered during certain times of the day while spaced out at other times. One key question was whether or not the spiders spaced out "optimally" to cooperate in prey capture. In this paper, we first show the derivation of the model where experimental data is used to determine key parameters within the model. We then use this model to test the success of prey capture under a variety of different spatial configurations for varying colony sizes to determine the best spatial configuration for prey capture.

Performing detailed simulations of plasma kinetic diffusion is a challenging task and currently requires the largest computational facilities in the world. The reason for this is that, the physics in a confined heated plasma occur on a broad range of temporal and spatial scales. It is therefore of interest to improve the computational algorithms together with the development of more powerful computational resources. Kinetic diffusion processes in plasmas are commonly simulated with the Monte Carlo method, where a discrete set of particles are sampled from a distribution function and advanced in a Lagrangian frame according to a set of stochastic differential equations. The Monte Carlo method introduces computational error in the form of statistical random noise produced by a finite number of particles (or markers) N and the error scales as αN−β where β = 1/2 for the standard Monte Carlo method. This requires a large number of simulated particles in order to obtain a sufficiently low numerical noise level. Therefore it is essential to use techniques that reduce the numerical noise. Such methods are commonly called variance reduction methods. In this thesis, we have developed new variance reduction methods with application to plasma kinetic diffusion. The methods are suitable for simulation of RF-heating and transport, but are not limited to these types of problems. We have derived a novel variance reduction method that minimizes the number of required particles from an optimization model. This implicitly reduces the variance when calculating the expected value of the distribution, since for a fixed error the  optimization model ensures that a minimal number of particles are needed. Techniques that reduce the noise by improving the order of convergence, have also been considered. Two different methods have been tested on a neutral beam injection scenario. The methods are the scrambled Brownian bridge method and a method here called the sorting and mixing method of L´ecot and Khettabi[1999]. Both methods converge faster than the standard Monte Carlo method for modest number of time steps, but fail to converge correctly for large number of time steps, a range required for detailed plasma kinetic simulations. Different techniques are discussed that have the potential of improving the convergence to this range of time steps. / QC 20120314

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This thesis is concerned with the mathematical analysis of electricity and carbon emission markets. We introduce a novel, versatile and tractable stochastic framework for the joint price formation of electricity spot prices and allowance certificates. In the proposed framework electricity and allowance prices are explained as functions of specific fundamental factors, such as the demand for electricity and the prices of the fuels used for its production. As a result, the proposed model very clearly captures the complex dependency of the modelled prices on the aforementioned fundamental factors. The allowance price is obtained as the solution to a coupled forward-backward stochastic differential equation. We provide a rigorous proof of the existence and uniqueness of a solution to this equation and analyse its behaviour using asymptotic techniques. The essence of the model for the electricity price is a carefully chosen and explicitly constructed function representing the supply curve in the electricity market. The model we propose accommodates most regulatory features that are commonly found in implementations of emissions trading systems and we analyse in detail the impact these features have on the prices of allowance certificates. Thereby we reveal a weakness in existing regulatory frameworks, which, in rare cases, can lead to allowance prices that do not conform with the conditions imposed by the regulator. We illustrate the applicability of our model to the pricing of derivative contracts, in particular clean spread options and numerically illustrate its ability to "see" relationships between the fundamental variables and the option contract, which are usually unobserved by other commonly used models in the literature. The results we obtain constitute flexible tools that help to efficiently evaluate the financial impact current or future implementations of emissions trading systems have on participants in these markets.

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.

This thesis uses a simple deterministic model represented by an ordinary di- fferential equation with two equilibrium points - depending on the initial state the illness either vanishes or persists forever. This model is expanded by adding some diffusion coefficients leading to different stochastic differential equations. They are analyzed to show how the choice of diffusion coefficients changes be- havior of the model in proximity of its equilibria and near the boundary of area with biological meaning. The theoretical results are than illustrated by computer simulations. 1

[pt] O presente trabalho tem como finalidade avaliar os diferentes modelos de mistura para o cálculo da combustão de reagentes pré- misturados utilizando a abordagem de Reator Parcialmente Misturado (PaSR). Os modelos de mistura considerados neste trabalho foram os modelos IEM estendido, Langevin e Langevin estendido. Investiga-se aqui o grau de mistura previsto por tais modelos e sua influência sobre as propriedades termoquímicas em um processo de combustão. A primeira parte deste trabalho consiste na apresentação e avaliação destes modelos de mistura, considerando-se um campo escalar inerte em presença de um campo turbulento homogêneo e isotrópico. Uma vez que estes modelos de mistura envolvem formulações do tipo estocástico, sua implementação foi realizada utilizando o método de Monte Carlo, mediante a utilização de esquemas numéricos adequados à resolução de equações diferenciais estocásticas. Assim, estuda-se a evolução da Função Densidade de Probabilidade (PDF) e das principais propriedades do campo escalar para cada modelo implementado. Os resultados obtidos também são comparados com simulação numérica direta e com resultados analáticos disponsáveis. Um ótimo acordo em termos qualitativos e quantitativos é obtido. A segunda parte deste trabalho utiliza estes modelos para o estudo numérico de um PaSR no qual são modelados os processos difusivos e reativos presentes durante a combustão. O PaSR é usado para avaliar a influência dos modelos de mistura nas propriedades termoquímicas da mistura em uma situação de combustão de tipo pré-misturada, que é modelada utilizando-se uma variável de progresso de uma reação. Os resultados obtidos com os diferentes modelos de mistura são comparados para diferentes regimes de funcionamento do PaSR, mostrando que, em situações de mistura rápida e reação intensa, os diferentes modelos apresentam resultados similares. Porém, nos casos de mistura lenta e reação moderada, discrepancias importantes são observadas entre os resultados dos modelos; as quais atingem até 65% para o valor médio da variável de progresso da reação. / [en] The present work evaluates several mixing models for the prediction of premixed combustion in a Partially Stirred Reactor (PaSR). The models considered in this work were the extended IEM, Langevin and extended Langevin models. The degree of mixing and its influence on the termochemical properties in a combustion process are investigated here. The first part of this work consists on the presentation and the assesment of these mixing models in which a single scalar field was considered in presence of a homogeneous and isotropic turbulent field. Since these mixing models involve stochastic terms, their implementation is performed by the Monte Carlo method using numerical schemes which solve the corresponding Stochastic Differential Equations (SDE). The evolution of the Probability Density Function (PDF) and the main properties for a single scalar field are studied for each mixing model. The numerical results are compared with Direct Numerical Simulation and available analytical results. Excellent qualitative and quantitative agreements are obtained. In the second part of this work, mixing models are used for numerical simulation of a PaSR where the diffusive and reactive processes occur. The PaSR is used to assess the mixing model influence on the termochemical properties of the mixture in a premixed combustion process, which is modeled using a reaction progress variable. The results obtained with the different mixing models are compared in several operating regimes of the PaSR, showing that when mixing is fast and reaction is intense, the different models lead to similar results. However, when mixing is slow and reaction is weak, important discrepancies are observed between the model results, which reach 65%, as far as the averaged reaction progress variable is concerned

Reatores tubulares de polimerização podem apresentar um perfil de velocidade bastante distorcido. Partindo desta observação, um modelo estocástico baseado no modelo de dispersão axial foi proposto para a representação matemática da fluidodinâmica de um reator tubular para produção de poliestireno. A equação diferencial foi obtida inserindo a aleatoriedade no parâmetro de dispersão, resultando na adição de um termo estocástico ao modelo capaz de simular as oscilações observadas experimentalmente. A equação diferencial estocástica foi discretizada e resolvida pelo método Euler-Maruyama de forma satisfatória. Uma função estimadora foi desenvolvida para a obtenção do parâmetro do termo estocástico e o parâmetro do termo determinístico foi calculado pelo método dos mínimos quadrados. Uma análise de convergência foi conduzida para determinar o número de elementos da discretização e o modelo foi validado através da comparação de trajetórias e de intervalos de confiança computacionais com dados experimentais. O resultado obtido foi satisfatório, o que auxilia na compreensão do comportamento fluidodinâmico complexo do reator estudado. / The velocity profile of polymerization tubular reactors may be very distorted. Based on this observation, a stochastic model based on the axial dispersion model was proposed for the mathematical representation of the fluid dynamics of a tubular reactor for polystyrene production. The differential equation was built by inserting randomness in the dipersion coefficient, which added a stochastic term to the model. This term was capable of simulating the experimentally observed fluctuations. The stochastic differential equation was discretized and solved by the Euler-Maruyama method adequately. An estimator function has been developed to calculate the parameter of the stochastic term, while the parameter of the deterministic term was estimated by a least squares method. A convergence analysis was carried out in order to determine the number of elements needed for the time discretization. The model was validated through comparisons of sample paths and computational confidence intervals with experimental data. The result was considered satisfactory, allowing a better understanding of the complex fluid dynamic behaviour of the analised reactor. Key-words: modelling, simulation, stochastic differential equation, polymerization tubular reactor, time residence distribution.

Der erste Teil der Arbeit beschreibt eine Methode, Auszahlungen zu bewerten, die einem auf dem Finanzmarkt nicht absicherbaren Risiken ausgesetzt sind. Im zweiten Teil berechnen wir den maximalen Nutzen und optimale Handelsstrategien auf unvollständigen Märkten mit Hilfe von stochastischen Rückwärtsgleichungen. Wir betrachten Händler, deren Einkommen einer externen Risikoquelle ausgesetzt sind. Diese vervollständigen den Markt, indem sie entweder einen Bond schaffen oder gegenseitig Verträge schliessen. Eine andere Moeglichkeit ist eine Anleihe, die von einer Versicherung herausgegeben wird. Die Risikoquellen, die wir in Betracht ziehen, können Versicherungs-, Wetter-oder Klimarisiko sein. Aktienpreise sind exogen gegeben. Wir berechnen Preise für die zusätzlichen Anlagen so dass Angebot und Nachfrage dafür gleich sind. Wir haben partielle Markträumung. Die Präferenzen der Händler sind durch erwarteten Nutzen gegeben. In Kapitel 2 bis Kapitel 4 haben die Händler exponentielle Nutzenfunktionen. Um den Gleichgewichtspreis zu finden, wenden wir stochastische Rückwärtsgleichungen an. In Kapitel 5 beschreiben wir ein Einperiodenmodell mit Nutzenfunktionen, die die Inada-Bedingungen erfüllen. Der zweite Teil dieser Arbeit beschäftigt sich mit dem robusten Nutzenmaximierungsproblem auf einem unvollständigen Finanzmarkt. Entweder das Wahrscheinlichkeitsmass oder die Koeffizienten des Aktienmarktes sind ungewiss. Die Lösung der Rückwärtsgleichung beschreibt die nutzenmaximierende Handelsstrategie und das Wahrscheinlichkeitsmass, das in der Auswertung des robusten Nutzens benutzt wird. Für die exponentielle Nutzenfunktion berechnen wir Nutzenindifferenzpreise. Ausserdem wenden wir diese Techniken auf die Maximierung des erwarteten Nutzens bezüglich eines festen Wahrscheinlichkeitsmasses an. Dafür betrachten wir abgeschlossene, im allgemeinen nicht konvexe zulässige Mengen für die Handelsstrategien. / The first part of the thesis proposes a method to find prices and hedging strategies for risky claims exposed to a risk factor that is not hedgeable on a financial market. In the second part we calculate the maximal utility and optimal trading strategies on incomplete markets using Backward Stochastic Differential Equations. We consider agents with incomes exposed to a non-hedgeable external source of risk by creating either a bond or by signing contracts. The sources of risk we think of may be insurance, weather or climate risk. Stock prices are seen as exogenuosly given. We calculate prices for the additional securities such that supply is equal to demand, the market clears partially. The preferences of the agents are described by expected utility. In Chapter 2 through Chapter 4 the agents use exponential utility functions, the model is placed in a Brownian filtration. In order to find the equilibrium price, we use Backward Stochastic Differential Equations. Chapter 5 provides a one--period model where the agents use utility functions satisfying the Inada condition. The second part of this thesis considers the robust utility maximization problem on an incomplete financial market. Either the probability measure or drift and volatility of the stock price process are uncertain. We apply a martingale argument and solve a saddle point problem. The solution of a Backward Stochastic Differential Equation describes the maximizing trading strategy as well as the probability measure that is used in the robust utility. We consider the exponential, the power and the logarithmic utility functions. For the exponential utility function we calculate utility indifference prices of not perfectly hedgeable claims. Finally, we maximize the expected utility with respect to a single probability measure. We apply a martingale argument and solve maximization problems. This allows us to consider closed, in general non--convex constraints on the values of trading strategies.