Thesis advisor: Nancy Rallis / The theory of Brownian motion is an integral part of statistics and probability, and it also has some of the most diverse applications found in any topic in mathematics. With extensions into fields as vast and different as economics, physics, and management science, Brownian motion has become one of the most studied mathematical phenomena of the late twentieth and early twenty-first centuries. Today, Brownian motion is mostly understood as a type of mathematical process called a stochastic process. The word "stochastic" actually stems from the Greek word for "I guess," implying that stochastic processes tend to produce uncertain results, and Brownian motion is no exception to this, though with the right models, probabilities can be assigned to certain outcomes and we can begin to understand these complicated processes. This work reaches to attain this goal with regard to Brownian motion, and in addition it explores several applications found in the aforementioned fields and beyond. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

This thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of random analytic polynomials. Building on the work of Li and Wei [28] we obtain precise asymptotics for the average number of zeros of this model. The primary tools used in this section are the famous Kac-Rice formula as well as classical methods in the asymptotic analysis of integrals such as the Laplace method. Continuing, we characterize several topological properties of this model of harmonic polynomials. In chapter 3 we obtain experimental results concerning the number of connected components of the orientation-reversing region as well as the geometry of the distribution of zeros. The tools used in this section are primarily Monte Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology software known as Perseus. While the results in this chapter are empirical rather than formal proofs, they lead to several enticing conjectures and open problems. Finally, in chapter 4 we address an industry problem in applied mathematics and machine learning. The analysis in this chapter implements similar techniques to those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or Control Area Network) is a network for allowing micro-controllers inside of vehicles to communicate with each other. We propose and demonstrate the e ectiveness of an algorithm for detecting malicious tra c using an approach that discovers and exploits the natural geometry of the CAN surface and its relationship to random walk Markov chains. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

by Shing-chiang Wong. / Bibliography: leaf 73 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986

Tang Fung Chu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 79-81). / Abstracts in English and Chinese. / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Multivariate Model and Bayesian Analysis of Stochastic Prior Information --- p.4 / Chapter 2.1 --- The Model --- p.4 / Chapter 2.2 --- Identification of the Model --- p.5 / Chapter 2.3 --- Bayesian Analysis of Stochastic Prior Information --- p.8 / Chapter 2.4 --- Computational Procedure --- p.10 / Chapter 2.4.1 --- Optimization Procedures --- p.11 / Chapter 2.4.2 --- Analytical Expressions --- p.12 / Chapter Chapter 3. --- Example and Simulation Study --- p.18 / Chapter 3.1 --- Example --- p.18 / Chapter 3.2 --- Simulation Study --- p.19 / Chapter 3.2.1 --- Designs --- p.20 / Chapter 3.2.2 --- Results --- p.23 / Chapter Chapter 4. --- Conclusion --- p.26 / Tables --- p.29 / References --- p.79

Yang, Nian. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 95-100). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Typescript, etc. / Digitized by Kansas Correctional Industries

Recent experiments have shown the fundamental role that random fluctuations play in many chemical systems in living cells, such as gene regulatory networks. Mathematical models are thus indispensable to describe such systems and to extract relevant biological information from experimental data. Recent decades have seen a considerable amount of modelling effort devoted to this task. However, current methodologies still present outstanding mathematical and computational hurdles. In particular, models which retain the discrete nature of particle numbers incur necessarily severe computational overheads, greatly complicating the tasks of characterising statistically the noise in cells and inferring parameters from data. In this thesis we study analytical approximations and inference methods for stochastic reaction dynamics. The chemical master equation is the accepted description of stochastic chemical reaction networks whenever spatial effects can be ignored. Unfortunately, for most systems no analytic solutions are known and stochastic simulations are computationally expensive, making analytic approximations appealing alternatives. In the case where spatial effects cannot be ignored, such systems are typically modelled by means of stochastic reaction-diffusion processes. As in the non-spatial case an analytic treatment is rarely possible and simulations quickly become infeasible. In particular, the calibration of models to data constitutes a fundamental unsolved problem. In the first part of this thesis we study two approximation methods of the chemical master equation; the chemical Langevin equation and moment closure approximations. The chemical Langevin equation approximates the discrete-valued process described by the chemical master equation by a continuous diffusion process. Despite being frequently used in the literature, it remains unclear how the boundary conditions behave under this transition from discrete to continuous variables. We show that this boundary problem results in the chemical Langevin equation being mathematically ill-defined if defined in real space due to the occurrence of square roots of negative expressions. We show that this problem can be avoided by extending the state space from real to complex variables. We prove that this approach gives rise to real-valued moments and thus admits a probabilistic interpretation. Numerical examples demonstrate better accuracy of the developed complex chemical Langevin equation than various real-valued implementations proposed in the literature. Moment closure approximations aim at directly approximating the moments of a process, rather then its distribution. The chemical master equation gives rise to an infinite system of ordinary differential equations for the moments of a process. Moment closure approximations close this infinite hierarchy of equations by expressing moments above a certain order in terms of lower order moments. This is an ad hoc approximation without any systematic justification, and the question arises if the resulting equations always lead to physically meaningful results. We find that this is indeed not always the case. Rather, moment closure approximations may give rise to diverging time trajectories or otherwise unphysical behaviour, such as negative mean values or unphysical oscillations. They thus fail to admit a probabilistic interpretation in these cases, and care is needed when using them to not draw wrong conclusions. In the second part of this work we consider systems where spatial effects have to be taken into account. In general, such stochastic reaction-diffusion processes are only defined in an algorithmic sense without any analytic description, and it is hence not even conceptually clear how to define likelihoods for experimental data for such processes. Calibration of such models to experimental data thus constitutes a highly non-trivial task. We derive here a novel inference method by establishing a basic relationship between stochastic reaction-diffusion processes and spatio-temporal Cox processes, two classes of models that were considered to be distinct to each other to this date. This novel connection naturally allows to compute approximate likelihoods and thus to perform inference tasks for stochastic reaction-diffusion processes. The accuracy and efficiency of this approach is demonstrated by means of several examples. Overall, this thesis advances the state of the art of modelling methods for stochastic reaction systems. It advances the understanding of several existing methods by elucidating fundamental limitations of these methods, and several novel approximation and inference methods are developed.

This thesis studies the impact of various fundamental frictions in the microstructure of financial markets. Specific market frictions we consider are latency in high-frequency trading, transaction costs arising from price impact or commissions, unhedgeable inventory risks due to stochastic volatility and time-varying liquidity costs. We explore the implications of each of these frictions in rigorous theoretical models from an investor's point of view and derive analytical expressions or efficient computational procedures for dynamic strategies. Specific methodologies in computing these policies include stochastic control theory, dynamic programming and tools from applied probability and stochastic processes. In the first chapter, we describe a theoretical model for the quantitative valuation of latency and its impact on the optimal dynamic trading strategy. Our model measures the trading frictions created by the presence of latency, by considering the optimal execution problem of a representative investor. Via a dynamic programming analysis, our model provides a closed-form expression for the cost of latency in terms of well-known parameters of the underlying asset. We implement our model by estimating the latency cost incurred by trading on a human time scale. Examining NYSE common stocks from 1995 to 2005 shows that median latency cost across our sample more than tripled during this time period. In the second chapter, we provide a highly tractable dynamic trading policy for portfolio choice problems with return predictability and transaction costs. Our rebalancing rule is a linear function of the return predicting factors and can be utilized in a wide spectrum of portfolio choice models with minimal assumptions. Linear rebalancing rules enable to compute exact and efficient formulations of portfolio choice models with linear constraints, proportional and nonlinear transaction costs, and quadratic utility function on the terminal wealth. We illustrate the implementation of the best linear rebalancing rule in the context of portfolio execution with positivity constraints in the presence of short-term predictability. We show that there exists a considerable performance gain in using linear rebalancing rules compared to static policies with shrinking horizon or a dynamic policy implied by the solution of the dynamic program without the constraints. Finally, in the last chapter, we propose a factor-based model that incorporates common factor shocks for the security returns. Under these realistic factor dynamics, we solve for the dynamic trading policy in the class of linear policies analytically. Our model can accommodate stochastic volatility and liquidity costs as a function of factor exposures. Calibrating our model with empirical data, we show that our trading policy achieves superior performance in the presence of common factor shocks.

This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes. The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.

Bryan Chi-ho Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 163-166). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background --- p.4 / Chapter 2.1 --- Constraint Satisfaction Problems --- p.4 / Chapter 2.2 --- Systematic Search --- p.5 / Chapter 2.3 --- Stochastic Search --- p.6 / Chapter 2.3.1 --- Overview --- p.6 / Chapter 2.3.2 --- GENET --- p.8 / Chapter 2.3.3 --- CSVC --- p.10 / Chapter 2.3.4 --- Adaptive Search --- p.12 / Chapter 2.4 --- Hybrid Approach --- p.13 / Chapter 3 --- Progressive Stochastic Search --- p.14 / Chapter 3.1 --- Progressive Stochastic Search --- p.14 / Chapter 3.1.1 --- Network Architecture --- p.15 / Chapter 3.1.2 --- Convergence Procedure --- p.16 / Chapter 3.1.3 --- An Illustrative Example --- p.21 / Chapter 3.2 --- Incremental Progressive Stochastic Search --- p.23 / Chapter 3.2.1 --- Network Architecture --- p.24 / Chapter 3.2.2 --- Convergence Procedure --- p.24 / Chapter 3.2.3 --- An Illustrative Example --- p.25 / Chapter 3.3 --- Heuristic Cluster Selection Strategy --- p.28 / Chapter 4 --- Experiments --- p.31 / Chapter 4.1 --- N-Queens Problems --- p.32 / Chapter 4.2 --- Permutation Generation Problems --- p.53 / Chapter 4.2.1 --- Increasing Permutation Problems --- p.54 / Chapter 4.2.2 --- Random Permutation Generation Problems --- p.75 / Chapter 4.3 --- Latin Squares and Quasigroup Completion Problems --- p.96 / Chapter 4.3.1 --- Latin Square Problems --- p.96 / Chapter 4.3.2 --- Quasigroup Completion Problems --- p.118 / Chapter 4.4 --- Random CSPs --- p.120 / Chapter 4.4.1 --- Tight Random CSPs --- p.139 / Chapter 4.4.2 --- Phase Transition Random CSPs --- p.156 / Chapter 5 --- Concluding Remarks --- p.159 / Chapter 5.1 --- Contributions --- p.159 / Chapter 5.2 --- Future Work --- p.161