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The concept of backward stochastic differential equation (BSDE) was initially brought up by Bismut when studying the stochastic optimal control problem. And it has been applied to describe various problems particularly to those in finance. After the fundamental work by Pardoux and Peng who proved the well-posedness of the nonlinear BSDE, the BSDE has been investigated intensively for both theoretical and practical purposes. In this thesis, we are concerned with a class of numerical methods for solving BSDEs, especially the one proposed by Zhao et al.. For this method, the convergence theory of the semi-discrete scheme (the scheme that discretizes the equation only in time) was already established, we shall further provide the analysis for the fully discrete scheme (the scheme that discretizes in both time and space). Moreover, using the BSDE as the adjoint equation, we shall construct the numerical method for solving the stochastic optimal control problem. We will discuss the situation when the control is deterministic as well as when the control is feedback.

In this thesis, we present three projects studying exactly solvable models in the KPZ universality class and one project studying a generalization of the SIR model from epidemiology. The first chapter gives an overview of the results and how they fit into the study of KPZ universality when applicable. Each of the following 4 chapters corresponds to a published or submitted article. In the first project, we study an oriented first passage percolation model for the evolution of a river delta. We show that at any fixed positive time, the width of a river delta of length L approaches a constant times L²/³ with Tracy-Widom GUE fluctuations of order L⁴/⁹. This result can be rephrased in terms of a particle system generalizing pushTASEP. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations. In the second project, we study n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a diffusive limit of the exactly solvable beta random walk in random environment. In the third project, we study a class of probability distributions on the six-vertex model, which originates from the higher spin vertex model. For these random six-vertex models we show that the behavior near their base is asymptotically described by the GUE-corners process. In the fourth project, we study a model for the spread of an epidemic. This model generalizes the classical SIR model to account for inhomogeneity in the infectiousness and susceptibility of individuals in the population. A first statement of this model is given in terms of infinitely many coupled differential equations. We show that solving these equations can be reduced to solving a one dimensional first order ODE, which is easy to solve numerically. We use the explicit form of this ODE to characterize the total number of people who are ever infected before the epidemic dies out. This model is not related to the KPZ universality class.

Storm induced flooding presents a multifaceted threat to coastal communities across the world.With climate change and sea level rise this danger is expected to increase. As coastal communities become exposed to more frequent and more severe flooding, the need for protective measures will increase. To know how to optimally protect against coastal flooding requires an understanding of future flood risk, storms, and storm surge. These are challenging to estimate due to many sources of uncertainty. In this thesis I present a methodology to forecast this future flood risk. I combine multiple computational, physics and statistical models to accurately describe the fluid dynamics of flooding, the cyclones that drive surge, and how climate change will influence these different components in the future. These computational models must be fast so that they can be embedded into an optimization framework that makes many evaluations. To find an optimal protective measure I employ stochastic and derivative free optimization methods. A complete study is conducted on New York City and optimal protective strategies are found for minimizing the total cost from storm surge subject to different budget constraints.

The thesis covers two fundamental topics that are important across the disciplines of operations research, statistics and even more broadly, namely stochastic optimization and uncertainty quantification, with the common theme to address both statistical accuracy and computational constraints. Here, statistical accuracy encompasses the precision of estimated solutions in stochastic optimization, as well as the tightness or reliability of confidence intervals. Computational concerns arise from rare events or expensive models, necessitating efficient sampling methods or computation procedures. In the first half of this thesis, we study stochastic optimization that involves rare events, which arises in various contexts including risk-averse decision-making and training of machine learning models. Because of the presence of rare events, crude Monte Carlo methods can be prohibitively inefficient, as it takes a sample size reciprocal to the rare-event probability to obtain valid statistical information about the rare-event. To address this issue, we investigate the use of importance sampling (IS) to reduce the required sample size. IS is commonly used to handle rare events, and the idea is to sample from an alternative distribution that hits the rare event more frequently and adjusts the estimator with a likelihood ratio to retain unbiasedness. While IS has been long studied, most of its literature focuses on estimation problems and methodologies to obtain good IS in these contexts. Contrary to these studies, the first half of this thesis provides a systematic study on the efficient use of IS in stochastic optimization. In Chapter 2, we propose an adaptive procedure that converts an efficient IS for gradient estimation to an efficient IS procedure for stochastic optimization. Then, in Chapter 3, we provide an efficient IS for gradient estimation, which serves as the input for the procedure in Chapter 2. In the second half of this thesis, we study uncertainty quantification in the sense of constructing a confidence interval (CI) for target model quantities or prediction. We are interested in the setting of expensive black-box models, which means that we are confined to using a low number of model runs, and we also lack the ability to obtain auxiliary model information such as gradients. In this case, a classical method is batching, which divides data into a few batches and then constructs a CI based on the batched estimates. Another method is the recently proposed cheap bootstrap that is constructed on a few resamples in a similar manner as batching. These methods could save computation since they do not need an accurate variability estimator which requires sufficient model evaluations to obtain. Instead, they cancel out the variability when constructing pivotal statistics, and thus obtain asymptotically valid t-distribution-based CIs with only few batches or resamples. The second half of this thesis studies several theoretical aspects of these computation-aware CI construction methods. In Chapter 4, we study the statistical optimality on CI tightness among various computation-aware CIs. Then, in Chapter 5, we study the higher-order coverage errors of batching methods. Finally, Chapter 6 is a related investigation on the higher-order coverage and correction of distributionally robust optimization (DRO) as another CI construction tool, which assumes an amount of analytical information on the model but bears similarity to Chapter 5 in terms of analysis techniques.

The present work uses statistical mechanics tools to investigate the dynamics of markets, prices, trades and wealth distribution. We studied the evolution of market dynamics in different stages of historical development by analyzing commodity prices from two distinct periods : ancient Babylon, and medieval and early modern England. We find that the first-digit distributrions of both Babylon and England commodity prices follow Benford's Law, indicating that the data represent empirical observations typically arising from a free market. Further, we find that the normalized prices of both Babylon and England agricultural commodities are characterized by stretched exponential distributions, and exhibit persistent correlations of a power law type over long periods of up to several centuries, in contrast to contemporary markets. Our findings suggest that similar market interactions may underlie the dynamics of ancient agricultural commodity prices, and that these interactions may remain stable across centuries. To further investigate the dynamics of markets, we present the analogy between transfers of money between individuals and the transfer of energy through particle collisions by means of the kinetic theory of gases. We introduce a theoretical framework of how micro rules of trading lead to the emergence of income and wealth distribution. Particularly, we study the effects of different types of distribution of savings/investments among individuals in a society and different welfare/subsidies redistribution policies. Results show that while considering savings propensities, the models approach empirical distributions of wealth quite well. The effect of redistribution better captures specific features of the distributions which earlier models failed to do. Moreover, the models still preserve the exponential decay observed in empirical income distributions reported by tax data and surveys. / by Natalia E. Romero. / Vita. / Thesis (Ph.D.)--Florida Atlantic University, 2012. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2012. Mode of access: World Wide Web.

In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class. The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation 𝜢(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation. In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log 𝑡)²/³ and (log log 𝑡)¹/³ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds 𝛼(log log 𝑡)²/³. This has relevance from the point of view of fractal geometry of the KPZ equation. We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length 𝑡 and any 𝑝 ⋲ (0,1), the point on the path which is 𝑝𝑡 distance away from the origin stays within a 𝑂(1) stochastic window around a random point 𝙈_𝑝,𝑡 that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around 𝙈_𝑝,𝑡 converges in law to an explicit random density function as 𝑡 → ∞ without any scaling. The limiting random density is proportional to 𝑒^{-𝓡(x)} where 𝓡(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length 𝑡, upon 𝑡²/³ superdiffusive scaling, is tight (as 𝑡 → ∞) in the space of 𝐶([0,1]) valued random variables. On the other hand, as 𝑡 → 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge. In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2𝑁 in a half-space with i.i.d.Gamma⁻¹(2𝛳) distributed bulk weights and i.i.d. Gamma⁻¹(𝛼+𝛳) distributed boundary weights for 𝛳 > 0 and 𝛼 > -𝛳. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when 𝛼 ≥ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by 𝑁²/³ and fluctuations scaled by 𝑁¹/³ is tight. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The 𝛼 ≥ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the 𝛼 < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within 𝑂(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

"January 2003" Includes bibliographical references (leaves 120-129) Presents performance analyses of dynamic, stochastic task scheduling policies for a real- time-communications system where tasks lose value as they are delayed in the system.

"January 2003" / Includes bibliographical references (leaves 120-129) / x, 129 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Presents performance analyses of dynamic, stochastic task scheduling policies for a real- time-communications system where tasks lose value as they are delayed in the system. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 2003

Orientadores: Fábio Violaro, Edmilson da Silva Morais / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-17T02:12:55Z (GMT). No. of bitstreams: 1 Barreira_RamiroRoqueAntunes_M.pdf: 2303475 bytes, checksum: 72e03fe8e41e9e440f2d4a266666763d (MD5) Previous issue date: 2010 / Resumo: A análise Mel-Cepstral Generalizada (MGC) corresponde a uma abordagem para estimação de envoltória espectral de fala que unifica as análises LPC, Mel-LPC, Cepstral e Mel-Cepstral. A forma funcional do modelo MGC varia continuamente com dois parâmetros reais ? e ?, possibilitando que o modelo assuma diferentes características. A flexibilidade oferecida pelo modelo MGC aliada à sua estabilidade e bom desempenho sob manipulação de parâmetros tem feito com que os parâmetros MGC sejam empregados com sucesso em codificação de fala e síntese de fala via HMM (Hidden Markov Models). O presente trabalho foca os aspectos matemáticos da análise MGC, abordando e demonstrando, em extensão, a formulação em seus vieses analítico e computacional para a solução do modelo. As propriedades e formulações básicas da análise MGC são tratadas na perspectiva do espectro mel-logarítmico generalizado. Propõe-se um método para a computação dos coeficientes MGC e Mel-Cepstrais que não envolve o uso de fórmulas recursivas de transformação em freqüência. As análises e experimentos relacionados ao método encontram-se em estágio inicial e devem ser completados no sentido de se identificar a relação ganho computacional × qualidade da representação. / Abstract: Mel-Generalized Cepstral analysis (MGC) is an approach for speech spectral envelope estimation that unifies LPC, Mel-LPC, Cepstral and Mel-Cepstral Analysis. The functional form of the MGC model varies continuously with the real parameters ? e ?, enabling the model to acquire different characteristics. The flexibility of MGC model associated with its stability and good performance under parameter manipulation have made MGC parameters to be successfully employed in speech codification and HMM speech synthesis. The present study focuses on mathematical aspects of MGC analysis, treating and proving, in a fairly extended way, analytical and computational formulation for model solution. MGC analysis properties and basic formulation are treated in melgeneralized logarithmic spectrum perspective. A method for the computation of MGC and Mel-Cepstral coefficients that do not require frequency transformation recursion formulas is proposed. Experiments and analysis concerning the method are in their initial stage and needs to be completed in the sense to identify computational × representation performances. / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica