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Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.

Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.

<pre>In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. </pre> <pre><br></pre> <pre>The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements.</pre> <pre><br></pre> <pre>The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition.</pre> <pre><br></pre> <pre>One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter.</pre> <pre><br></pre> <pre>Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.</pre>

<div>The dissertation consists of two research topics.</div><div><br></div><div>The first research direction is to study stability of functional and geometric inequalities. Stability problem is to estimate the deficit of a functional or geometric inequality in terms of the distance from the class of optimizers or a functional that identifies the optimizers. In particular, we investigate the logarithmic Sobolev inequality, the Beckner-Hirschman inequality (the entropic uncertainty principle), and isoperimetric type inequalities for the expected lifetime of Brownian motion. </div><div><br></div><div>The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. We extend the Hardy-Littlewood-Sobolev inequality to symmetric Markov semigroups. To this end, we construct a stochastic representation of the fractional integral using the background radiation process. The inequality follows from a new inequality for the fractional Littlewood-Paley square function. We also prove the Hardy-Stein identity for non-symmetric pure jump Levy processes and the L^p boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump Levy processes. The proof is based on Ito's formula for general jump processes and the symmetrization of Levy processes. <br></div>

A useful measure of quality of a global optimisation algorithm such as simulated annealing is the length of time it must be run to reach a global optimum within a certain accuracy. Such a performance measure assists in choosing and tuning algorithms. This thesis proposes an approach to obtaining such a measure through successive approximation of a generic stochastic global optimisation algorithm with a sequence of stochastic processes culminating in backtracking adaptive search. The overall approach is to approximate the progress of an optimisation algorithm with that of a model process, backtracking adaptive search. The known convergence rate of the model then provides an estimator of the unknown convergence rate of the original algorithm. Parameters specifying this model are chosen based on observation of the optimisation algorithm. The optimisation algorithm may first be approximated with a time-inhomogeneous Markovian process defined on the problem range. The distribution of the number of iterations to convergence for this averaged range process is shown to be identical with that of the original process. This process is itself approximated by a time-homogeneous Markov process in the range, the asymptotic averaged range process. This approximation is defined for all Markovian optimisation algorithms and a weak condition under which its convergence time closely matches that of the original algorithm is developed. The asymptotic averaged range process is of the same form as backtracking adaptive search, the final stage of approximation. Backtracking adaptive search is an optimisation algorithm which generalises pure adaptive search and hesitant adaptive search. In this thesis the distribution of the number of iterations for which the algorithm runs in order to reach a sufficiently extreme objective function level is derived. Several examples of backtracking adaptive search on finite problems are also presented, including special cases that have received attention in the literature. Computational results of the entire approximation framework are reported for several examples. The method can be applied to any optimisation algorithm to obtain an estimate of the time required to obtain solutions of a certain quality. Directions for further work in order to improve the accuracy of such estimates are also indicated.

Probability theory is the study of random phenomena. Many dynamical systems with random influence, in nature or artificial complex systems, are better modeled by equations incorporating the intrinsic stochasticity involved. In probability theory, stochastic partial differential equations (SPDEs) generalize partial differential equations through random force terms and coefficients, while stochastic differential equations (SDEs) generalize ordinary differential equations. They are both abound in models involving Brownian motion throughout science, engineering and economics. However, Brownian motion is just one example of a random noisy input. The goal of this thesis is to make contributions in the study and applications of stochastic dynamical systems involving a wider variety of stochastic processes and noises. This is achieved by considering different models arising out of applications in thermal engineering, population dynamics and mathematical finance.<br><div><br></div><div>1. Power-type non-linearities in SDEs with rough noise: We consider a noisy differential equation driven by a rough noise that could be a fractional Brownian motion, a generalization of Brownian motion, while the equation's coefficient behaves like a power function. These coefficients are interesting because of their relation to classical population dynamics models, while their analysis is particularly challenging because of the intrinsic singularities. Two different methods are used to construct solutions: (i) In the one-dimensional case, a well-known transformation is used; (ii) For multidimensional situations, we find and quantify an improved regularity structure of the solution as it approaches the origin. Our research is the first successful analysis of the system described under a truly rough noise context. We find that the system is well-defined and yields non-unique solutions. In addition, the solutions possess the same roughness as that of the noise.<br></div><div><br></div><div>2. Parabolic Anderson model in rough environment: The parabolic Anderson model is one of the most interesting and challenging SPDEs used to model varied physical phenomena. Its original motivation involved bound states for electrons in crystals with impurities. It also provides a model for the growth of magnetic field in young stars and has an interpretation as a population growth model. The model can be expressed as a stochastic heat equation with additional multiplicative noise. This noise is traditionally a generalized derivative of Brownian motion. Here we consider a one dimensional parabolic Anderson model which is continuous in space and includes a more general rough noise. We first show that the equation admits a solution and that it is unique under some regularity assumptions on the initial condition. In addition, we show that it can be represented using the Feynman-Kac formula, thus providing a connection with the SPDE and a stochastic process, in this case a Brownian motion. The bulk of our study is devoted to explore the large time behavior of the solution, and we provide an explicit formula for the asymptotic behavior of the logarithm of the solution.<br></div><div><br></div><div>3. Heat conduction in semiconductors: Standard heat flow, at a macroscopic level, is modeled by the random erratic movements of Brownian motions starting at the source of heat. However, this diffusive nature of heat flow predicted by Brownian motion is not observed in certain materials (semiconductors, dielectric solids) over short length and time scales. The thermal transport in these materials is more akin to a super-diffusive heat flow, and necessitates the need for processes beyond Brownian motion to capture this heavy tailed behavior. In this context, we propose the use of a well-defined Lévy process, the so-called relativistic stable process to better model the observed phenomenon. This process captures the observed heat dynamics at short length-time scales and is also closely related to the relativistic Schrödinger operator. In addition, it serves as a good candidate for explaining the usual diffusive nature of heat flow under large length-time regimes. The goal is to verify our model against experimental data, retrieve the best parameters of the process and discuss their connections to material thermal properties.<br></div><div><br></div><div>4. Bond-pricing under partial information: We study an information asymmetry problem in a bond market. Especially we derive bond price dynamics of traders with different levels of information. We allow all information processes as well as the short rate to have jumps in their sample paths, thus representing more dramatic movements. In addition we allow the short rate to be modulated by all information processes in addition to having instantaneous feedbacks from the current levels of itself. A fully informed trader observes all information which affects the bond price while a partially informed trader observes only a part of it. We first obtain the bond price dynamic under the full information, and also derive the bond price of the partially informed trader using Bayesian filtering method. The key step is to perform a change of measure so that the dynamic under the new measure becomes computationally efficient.</div>