Modelling of volcanic ashfall has been attempted by volcanologists but very little work has been done by mathematicians. In this thesis we show that mathematical models can accurately describe the distribution of particulate materials that fall to the ground following an eruption. We also report on the development and analysis of mathematical models to calculate the ash concentration in the atmosphere during ashfall after eruptions. Some of these models have analytical solutions. The mathematical models reported on in this thesis not only describe the distribution of ashfall on the ground but are also able to take into account the effect of variation of wind direction with elevation. In order to model the complexity of the atmospheric flow, the atmosphere is divided into horizontal layers. Each layer moves steadily and parallel to the ground: the wind velocity components, particle settling speed and dispersion coefficients are assumed constant within each layer but may differ from layer to layer. This allows for elevation-dependent wind and turbulence profiles, as well as changing particle settling speeds, the last allowing the effects of the agglomeration of particles to be taken into account.

The dissertation consists of two parts. The first part (Chapter 1 to 4) is on some contributions to the development of a non-linear analysis on the quintessential fractal set Sierpinski gasket and its probabilistic interpretation. The second part (Chapter 5) is on the asymptotic tail decays for suprema of stochastic processes satisfying certain conditional increment controls. Chapters 1, 2 and 3 are devoted to the establishment of a theory of backward problems for non-linear stochastic differential equations on the gasket, and to derive a probabilistic representation to some parabolic type partial differential equations on the gasket. In Chapter 2, using the theory of Markov processes, we derive the existence and uniqueness of solutions to backward stochastic differential equations driven by Brownian motion on the Sierpinski gasket, for which the major technical difficulty is the exponential integrability of quadratic processes of martingale additive functionals. A Feynman-Kac type representation is obtained as an application. In Chapter 3, we study the stochastic optimal control problems for which the system uncertainties come from Brownian motion on the gasket, and derive a stochastic maximum principle. It turns out that the necessary condition for optimal control problems on the gasket consists of two equations, in contrast to the classical result on ℝ^d, where the necessary condition is given by a single equation. The materials in Chapter 2 are based on a joint work with Zhongmin Qian (referenced in Chapter 2). Chapter 4 is devoted to the analytic study of some parabolic PDEs on the gasket. Using a new type of Sobolev inequality which involves singular measures developed in Section 4.2, we establish the existence and uniqueness of solutions to these PDEs, and derive the space-time regularity for solutions. As an interesting application of the results in Chapter 4 and the probabilistic representation developed in Chapter 2, we further study Burgers equations on the gasket, to which the space-time regularity for solutions is deduced. The materials in Chapter 4 are based on a joint work with Zhongmin Qian (referenced in Chapter 4). In Chapter 5, we consider a class of continuous stochastic processes which satisfy the conditional increment control condition. Typical examples include continuous martingales, fractional Brownian motions, and diffusions governed by SDEs. For such processes, we establish a Doob type maximal inequality. Under additional assumptions on the tail decays of their marginal distributions, we derive an estimate for the tail decay of the suprema (Theorem 5.3.2), which states that the suprema decays in a manner similar to the margins of the processes. In Section 5.4, as an application of Theorem 5.3.2, we derive the existence of strong solutions to a class of SDEs. The materials in this chapter is based on the work [44] by the author (Section 5.2 and Section 5.3) and an ongoing joint project with Guangyu Xi (Section 5.4).

This dissertation studies the pricing of Outside barrier call options, when their activation starts at a hitting time. The pricing of Outside barrier options when their activation starts at time zero, and the pricing of standard barrier options when their activation starts at a hitting time of a pre speci ed barrier level, have been studied previously (see [21], [24]). The new work that this dissertation will do is to price Outside barrier call options, where they will be activated when the triggering asset crosses or hits a pre speci ed barrier level, and also the pricing of Outside barrier call options where they will be activated when the triggering asset crosses or hits a sequence of two pre specifed barrier levels. Closed form solutions are derived using Girsanov's theorem and the re ection principle. Existing results are derived from the new results, and properties of the new results are illustrated numerically and discussed. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Orientador: Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T12:34:53Z (GMT). No. of bitstreams: 1 Felix_LucianoVianna_M.pdf: 566321 bytes, checksum: f717034fada0c65f1b886ba7bd821902 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho estudamos a geometria dos caminhos em grupos de Lie usando a exponencial estocástica e o logaritmo estocástico. Apresentamos as construções geométricas do espaço tangente, uma métrica e uma conexão natural as caminhos em grupos de Lie. Finalmente apresentamos uma situação em que essa conexão é Levi-Civita e outra que não é / Abstract: In this work, we study the path geometry in Lie groups using the stochastic exponential and the stochastic logarithm. We show the geometric constructions of tangent space, one metric and one natural conection of Lie groups valued path. Finelly we show one situation that this conection is Levi-Civita and another one that is not / Mestrado / Geometria / Mestre em Matemática

Orientador: Pablo Siqueira Meirelles / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-20T08:25:52Z (GMT). No. of bitstreams: 1 Piveta_Alessandro_M.pdf: 20999298 bytes, checksum: 60bf64ff246a3515b0455a01ccb8e817 (MD5) Previous issue date: 2012 / Resumo: No desenvolvimento de projetos mecânicos é comum o uso de valores determinísticos para definir parâmetros que descrevem o sistema, atribuindo elevados coeficientes de segurança a fim de contornar o problema de incertezas relacionadas aos valores das cargas externas aplicadas, da variabilidade das propriedades dos materiais, das condições ambientais, entre outros fatores que possam afetar o desempenho do sistema. Atualmente, com o desenvolvimento de alguns métodos, pode-se introduzir modelos estocásticos que consideram estas incertezas e variações, deixando assim o modelo mais próximo da realidade. Este trabalho analisa o efeito da variabilidade simultânea de determinados parâmetros no desempenho de dois tipos de suspensão: convencional e hidropneumática. Utilizou-se um modelo matemático que representa um veículo completo com sete graus de liberdade implementado no software Matlab®. É desenvolvida a formulação para modelar a suspensão hidropneumática considerando que o gás contido no acumulador sofre uma mudança de estado politrópico. Inicialmente são apresentadas as respostas dos modelos determinísticos devido a excitações transientes representadas por funções que descrevem uma lombada e um degrau com diferentes dimensões. Em seguida, é realizada a análise de sensibilidade da rigidez hidropneumática com o intuito de descobrir quais parâmetros possuem maior influência sobre este sistema. O Método de Monte Carlo é utilizado como solver estocástico nas simulações, tendo como objetivo avaliar a influência das variações dos parâmetros sobre a dinâmica vertical do veículo. A distribuição de probabilidade escolhida para cada parâmetro é a função gama. Por último, são analisadas as respostas resultantes dos modelos probabilísticos. As saídas avaliadas são: aceleração vertical do centro de gravidade, razão de amortecimento, valor da força de contato entre pneu e via e ângulo de rolagem da massa suspensa / Abstract: Deterministic values are largely used in mechanical design to define the parameters which describe the system. Large coefficients of safety are used because of the uncertainties related to these parameters. There are uncertainties related to the external loadings, to the variability of the material properties and to the environmental conditions, among other important parameters which affect the performance of the system. In the last decade, stochastic models have been used to consider these uncertainties and variations. The present work analyzes the effect of the simultaneous variability of certain parameters on the performance of the conventional and the hydropneumatic suspension of automotive vehicles. A mathematical model of a full vehicle with seven degrees of freedom has been used. This model was implemented in the software Matlab®. In the present formulation of the hydropneumatic suspension, it is considered that the gas contained in the accumulator undergoes a polytrophic process. First, the deterministic model of the vehicle is subjected to two transient excitations, namely a function describing a bump and a function describing a step with different dimensions. Then, an analysis of sensibility of the stiffness of the hydropneumatic suspension is performed, which aims to identify which parameters are the most influential on that system. Monte Carlo method is used as the stochastic solver throughout the simulations which study the influence of the parameter variability on the vertical dynamics of the vehicle. The Gamma function is used as the probability density function for each parameter. Finally, the response of the probabilistic models is studied. This work investigates the acceleration of the center of gravity, damping factor, pitch angle of the suspended mass and contact force between the tires and the road / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

Orientador: Paulo Regis Caron Ruffino / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T22:03:50Z (GMT). No. of bitstreams: 1 GonzalesGargate_IvanItalo_D.pdf: 724522 bytes, checksum: bb313a00b360e7bea2a2411b759c7389 (MD5) Previous issue date: 2012 / Resumo: Nesta tese, estudamos um princípio de médias em equações diferenciais estocásticas sobre variedades folheadas com folhas compactas. Começaremos introduzindo o princípio de médias sobre equações diferenciais ordinárias reais. A título de comparação vamos rever conceitos básicos de variedade simplética com a finalidade de comparar/estender os resultados obtidos por Xue-Mei Li sobre um princípio de médias para um sistema Hamiltoniano estocástico completamente integrável. Nosso principal resultado é generalizar estas idéias para o caso de uma variedade M = (-a; a)n x N, onde N é uma variedade compacta sem bordo. Em particular mostraremos nossos resultados para o caso que a folheação é gerada por uma submersão de M sobre Rn. Finalmente apresentamos alguns exemplos / Abstract: In this thesis, we study the averaging principle for stochastic differential equations on foliated manifolds with compact leaves. We begin by introducing the averaging principle over real ordinary differential equations. For comparison we will review basic concepts of symplectic manifold in order to compare/extend the results obtained by Xue-Mei Li about a averaging principle for a completely integrable stochastic Hamiltonian system. Our main result is to generalize these ideas to the case of a manifold M = (-a; a)n x N, where N is a compact manifold without boundary. In particular our results show for the case that foliation is generated by an submersion of M over Rn. Finally we present some examples / Doutorado / Matematica / Doutor em Matemática

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:49:45Z (GMT). No. of bitstreams: 1 SilvaJunior_RinaldoVieirada_M.pdf: 1586291 bytes, checksum: 8d01bdf434ecba2fb62a57725c46dd4a (MD5) Previous issue date: 2005 / Resumo: Nesta dissertação fizemos um estudo da teoria de difusão em variedades Finsler, onde abor-damos o transporte paralelo estocástico, desenvolvimento estocástico de Cartan e Movimento Browniano. O objetivo principal é obter uma descrição mais geométrica dos objetos citados acima ainda que por enquanto em coordenadas locais e assim termos um paralelo entre o cálculo estocástico em variedades Riemannianas e variedades Finsler / Abstract: In this work we study diffusion theory in Finsler manifolds. It includes the stochastic par-allel transport, stochastic Cartan development and Brownian motion. The main objective is to provide a geometric description of the objects mentioned and 50 to draw a compari-50n between stochastic calculus in Riemannian manifolds and stochastic calculus in Finsler manifolds / Mestrado / Matematica / Mestre em Matemática

Cette thèse développe une approche trajectorielle pour la modélisation des marchés financiers en temps continu, sans faire appel à des hypothèses probabilistes ou à des modèles stochastiques. À l'aide du calcul fonctionnel non-anticipatif, nous identifions une classe spéciale de stratégies de trading que nous prouvons être auto-finançantes, selon une notion trajectorielle introduite dans cette thèse, et dont le gain peut être calculé trajectoire par trajectoire comme limite de sommes de Riemann. Avec ces outils, nous proposons un cadre analytique pour analyser la performance et la robustesse de stratégies de couverture dynamique de produits dérivés path-dependent sur en ensemble de scénarios. Ce cadre ne demande aucune hypothèse probabiliste sur la dynamique du processus sous-jacent. Il généralise donc les résultats précédents sur la robustesse de stratégies de couverture dans des modèles de diffusion. Nous obtenons une formule explicite pour l'erreur de couverture dans chaque scénario et nous fournissons des conditions suffisantes qui impliquent la robustesse de la couverture delta-neutre. Nous montrons que la robustesse peut être obtenue dans un ensemble ample de modèles de prix de martingale exponentielle de carré intégrable, avec une condition de convexité verticale sur le payoff. Nous remarquons que les discontinuités de la trajectoire de prix détériorent la performance de la couverture. Le dernier chapitre, indépendant du reste de la thèse, est une étude en collaboration avec Andrea Pascucci et Stefano Pagliarani, où nous proposons une nouvelle méthode pour l'approximation analytique dans des modèles à volatilité locale avec des sauts de type Lévy. / This thesis develops a mathematical framework for the analysis of continuous-time trading strategies which, in contrast to the classical setting of continuous-time finance, does not rely on stochastic integrals or other probabilistic notions.Using the `non-anticipative functional calculus', we first develop a pathwise definition of the gain process for a large class of continuous-time trading strategies which includes delta-hedging strategies, as well as a pathwise definition of the self-financing condition. Using these concepts, we propose a framework for analyzing the performance and robustness of delta-hedging strategies for path-dependent derivatives across a given set of scenarios. Our setting allows for general path-dependent payoffs and does not require any probabilistic assumption on the dynamics of the underlying asset, thereby extending previous results on robustness of hedging strategies in the setting of diffusion models. We obtain a pathwise formula for the hedging error for a general path-dependent derivative and provide sufficient conditions ensuring the robustness of the delta-hedge. We show in particular that robust hedges may be obtained in a large class of continuous exponential martingale models under a vertical convexity condition on the payoff functional. We also show that discontinuities in the underlying asset always deteriorate the hedging performance. These results are applied to the case of Asian options and barrier options. The last chapter, independent of the rest of the thesis, proposes a novel method, jointly developed with Andrea Pascucci and Stefano Pagliarani, for analytical approximations in local volatility models with Lévy jumps.

<div>Complex systems are often characterized by a large collection of components interacting in nontrivial ways. Self-organization among these individual components often leads to emergence of a macroscopic structure that is neither completely regular nor completely random. In order to understand what we observe at a macroscopic scale, conceptual, mathematical, and computational tools are required for modeling and analyzing these interactions. A principled approach to understand these complex systems (and the processes that give rise to them) is to formulate generative models and infer their parameters from given data that is typically stored in the form of networks (or graphs). The increasing availability of network data from a wide variety of sources, such as the Internet, online social networks, collaboration networks, biological networks, etc., has fueled the rapid development of network science. </div><div><br></div><div>A variety of generative models have been designed to synthesize networks having specific properties (such as power law degree distributions, small-worldness, etc.), but the structural richness of real-world network data calls for researchers to posit new models that are capable of keeping pace with the empirical observations about the topological properties of real networks. The mechanistic approach to modeling networks aims to identify putative mechanisms that can explain the dependence, diversity, and heterogeneity in the interactions responsible for creating the topology of an observed network. A successful mechanistic model can highlight the principles by which a network is organized and potentially uncover the mechanisms by which it grows and develops. While it is difficult to intuit appropriate mechanisms for network formation, machine learning and evolutionary algorithms can be used to automatically infer appropriate network generation mechanisms from the observed network structure.</div><div><br></div><div>Building on these philosophical foundations and a series of (not new) observations based on first principles, we extrapolate an action-based framework that creates a compact probabilistic model for synthesizing real-world networks. Our action-based perspective assumes that the generative process is composed of two main components: (1) a set of actions that expresses link formation potential using different strategies capturing the collective behavior of nodes, and (2) an algorithmic environment that provides opportunities for nodes to create links. Optimization and machine learning methods are used to learn an appropriate low-dimensional action-based representation for an observed network in the form of a row stochastic matrix, which can subsequently be used for simulating the system at various scales. We also show that in addition to being practically relevant, the proposed model is relatively exchangeable up to relabeling of the node-types. </div><div><br></div><div>Such a model can facilitate handling many of the challenges of understanding real data, including accounting for noise and missing values, and connecting theory with data by providing interpretable results. To demonstrate the practicality of the action-based model, we decided to utilize the model within domain-specific contexts. We used the model as a centralized approach for designing resilient supply chain networks while incorporating appropriate constraints, a rare feature of most network models. Similarly, a new variant of the action-based model was used for understanding the relationship between the structural organization of human brains and the cognitive ability of subjects. Finally, our analysis of the ability of state-of-the-art network models to replicate the expected topological variations in network populations highlighted the need for rethinking the way we evaluate the goodness-of-fit of new and existing network models, thus exposing significant gaps in the literature.</div>

<p>This dissertation investigates the use of sampling methods for solving stochastic optimization problems using iterative algorithms. Two sampling paradigms are considered: (i) adaptive sampling, where, before each iterate update, the sample size for estimating the objective function and the gradient is adaptively chosen; and (ii) retrospective approximation (RA), where, iterate updates are performed using a chosen fixed sample size for as long as progress is deemed statistically significant, at which time the sample size is increased. We investigate adaptive sampling within the context of a trust-region framework for solving stochastic optimization problems in $\mathbb{R}^d$, and retrospective approximation within the broader context of solving stochastic optimization problems on a Hilbert space. In the first part of the dissertation, we propose Adaptive Sampling Trust-Region Optimization (ASTRO), a class of derivative-based stochastic trust-region (TR) algorithms developed to solve smooth stochastic unconstrained optimization problems in $\mathbb{R}^{d}$ where the objective function and its gradient are observable only through a noisy oracle or using a large dataset. Efficiency in ASTRO stems from two key aspects: (i) adaptive sampling to ensure that the objective function and its gradient are sampled only to the extent needed, so that small sample sizes are chosen when the iterates are far from a critical point and large sample sizes are chosen when iterates are near a critical point; and (ii) quasi-Newton Hessian updates using BFGS. We prove three main results for ASTRO and for general stochastic trust-region methods that estimate function and gradient values adaptively, using sample sizes that are stopping times with respect to the sigma algebra of the generated observations. The first asserts strong consistency when the adaptive sample sizes have a mild logarithmic lower bound, assuming that the oracle errors are light-tailed. The second and third results characterize the iteration and oracle complexities in terms of certain risk functions. Specifically, the second result asserts that the best achievable $\mathcal{O}(\epsilon^{-1})$ iteration complexity (of squared gradient norm) is attained when the total relative risk associated with the adaptive sample size sequence is finite; and the third result characterizes the corresponding oracle complexity in terms of the total generalized risk associated with the adaptive sample size sequence. We report encouraging numerical results in certain settings. In the second part of this dissertation, we consider the use of RA as an alternate adaptive sampling paradigm to solve smooth stochastic constrained optimization problems in infinite-dimensional Hilbert spaces. RA generates a sequence of subsampled deterministic infinite-dimensional problems that are approximately solved within a dynamic error tolerance. The bottleneck in RA becomes solving this sequence of problems efficiently. To this end, we propose a progressive subspace expansion (PSE) framework to solve smooth deterministic optimization problems in infinite-dimensional Hilbert spaces with a TR Sequential Quadratic Programming (SQP) solver. The infinite-dimensional optimization problem is discretized, and a sequence of finite-dimensional problems are solved where the problem dimension is progressively increased. Additionally, (i) we solve this sequence of finite-dimensional problems only to the extent necessary, i.e., we spend just enough computational work needed to solve each problem within a dynamic error tolerance, and (ii) we use the solution of the current optimization problem as the initial guess for the subsequent problem. We prove two main results for PSE. The first assesses convergence to a first-order critical point of a subsequence of iterates generated by the PSE TR-SQP algorithm. The second characterizes the relationship between the error tolerance and the problem dimension, and provides an oracle complexity result for the total amount of computational work incurred by PSE. This amount of computational work is closely connected to three quantities: the convergence rate of the finite-dimensional spaces to the infinite-dimensional space, the rate of increase of the cost of making oracle calls in finite-dimensional spaces, and the convergence rate of the solution method used. We also show encouraging numerical results on an optimal control problem supporting our theoretical findings.</p> <p>  </p>