This thesis is concerned with quantifying the dynamical role of stochasticity in models of recurrent epidemics. Although the simulation of stochastic models can accurately capture the qualitative epidemic patterns of childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns. The novel aspect of this thesis is the use of analytic methods to quantify the results from simulations. All the models are formulated as continuous time Markov processes, the temporal evolutions of which is described by a master equation. This is expanded in the inverse system size, which decomposes the full stochastic dynamics into a macroscopic part, described by deterministic equations, plus a stochastic fluctuating part. The first part examines the inclusion of non-exponential latent and infectious periods into the the standard susceptible-infectious-recovered model. The method of stages is used to formulate the problem as a Markov process and thus derive a power spectrum for the stochastic oscillations. This model is used to understand the dynamics of whooping cough, which we show to be the mixture of an annual limit cycle plus resonant stochastic oscillations. This limit cycle is generated by the time-dependent external forcing, but we show that the spectrum is close to that predicted by the unforced model. It is demonstrated that adding distributed infectious periods only changes the frequency and amplitude of the stochastic oscillations---the basic mechanisms remain the same. In the final part of this thesis, the effect of seasonal forcing is studied with an analysis of the full time-dependent master equation. The comprehensive nature of this approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in measles epidemics. In the pre-vaccination regime the dynamics are dominated by a period doubling bifurcation, which leads to large biennial oscillations in the deterministic dynamics. Vaccination is shown to move the system away from the biennial limit cycle and into a region where there is an annual limit cycle and stochastic oscillations, similar to whooping cough. Finite size effects are investigated and found to be of considerable importance for measles dynamics, especially in the biennial regime.

M.Ing. (Electrical and Electronic Engineering) / This treatise presents an investigation into the application of multivariable frequency domain techniques in the modelling and control of a helicopter aircraft in forward flight. The presentation is structured in the following sectioned format: I Hypotheses are stated which deal with the use of linear, multivariable, frequency domain theory in the modelling and control of helicopter aircraft. II The stated hypotheses are investigated by the application of relevant theories and techniques to a reference case plant - a single rotor helicopter in forward flight. III Conclusions drawn from the results are used to assess the validity of the hypotheses. The subject matter of the presentation may be summarized as follows: The hypotheses are initially placed in perspective by a discussion of the incentives for their formulation. In essence, the hypotheses state that helicopter dynamics, in a multivariable systems characterization, can be modelled and an appropriate flight control system designed by the use of linear frequency domain theory. The plant in reference to which the hypotheses are investigated is a single rotor utility helicopter - the Aerospatiale Alouette III. A single flight condition - a typical cruising condition - is considered. A comprehensive, nonlinear digital computer simulation of the aircraft is used as a substitute for the actual plant in the execution of the modelling and control design processes. The plant is modelled in terms of a linear model structure, in the form of the frequency response function, by linearization of its highly nonlinear dynamics about an operating point (datum flight condition). The frequency response function model parameters are identified by power spectral density analysis procedures. This method, based on random signal excitation of the plant, provides a valuable quantitative measure of the accuracy of the linearization performed in the identification. The measure, the coherence function, is used as a criterion for the robustness required of a control system of which the design is based on a linear model of a nonlinear plant.

This study gives a phenomenologically based stochastic model of space -time rainfall. Specifically, two random variables on the spatial rainfall, e.g. the cumulative rainfall within a season and the maximum cumulative rainfall per rainfall event within a season are considered. An approach is given to determine the cumulative distribution function (c.d.f.) of the cumulative rainfall per event, based on a particular random structure of space -time rainfall. Then the first two moments of the cumulative seasonal rainfall are derived based on a stochastic dependence between the cumulative rainfall per event and the number of rainfall events within a season. This stochastic dependence is important in the context of the spatial rainfall process. A theorem is then proved on the rate of convergence of the exact c.d.f. of the seasonal cumulative rainfall up to the ith year, i > 1, to its limiting c.d.f. Use of the limiting c.d.f. of the maximum cumulative rainfall per rainfall event up to the ith year within a season is given in the context of determination of the 'design rainfall'. Such information is useful in the design of hydraulic structures. Special mathematical applications of the general theory are developed from a combination of empirical and phenomenological based assumptions. A numerical application of this approach is demonstrated on the Atterbury watershed in the Southwestern United States.

This dissertation provides a brief theoretical introduction to the Malliavin calculus leading to a particular application in finance. The Malliavin calculus concepts are used to aid in the simulation of the Greeks for financial contingent claims. Particular focus is placed on creating efficiency in the more exotic type option simulations, where no closed solution pricing formulae exist. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.

Models governing stochasticity in various systems are typically calibrated from data, therefore are subject to statistical errors/uncertainties which can lead to inferior decision making. This thesis develops statistically and computationally efficient data-driven methods for problems in stochastic simulation and optimization to quantify and hedge impacts of these uncertainties. The first half of the thesis focuses on efficient methods for tackling input uncertainty which refers to the simulation output variability arising from the statistical noise in specifying the input models. Due to the convolution of the simulation noise and the input noise, existing bootstrap approaches consist of a two-layer sampling and typically require substantial simulation effort. Chapter 2 investigates a subsampling framework to reduce the required effort, by leveraging the form of the variance and its estimation error in terms of the data size and the sampling requirement in each layer. We show how the total required effort is reduced, and explicitly identify the procedural specifications in our framework that guarantee relative consistency in the estimation, and the corresponding optimal simulation budget allocations. In Chapter 3 we study an optimization-based approach to construct confidence intervals for simulation outputs under input uncertainty. This approach computes confidence bounds from simulation runs driven by probability weights defined on the data, which are obtained from solving optimization problems under suitably posited averaged divergence constraints. We illustrate how this approach offers benefits in computational efficiency and finite-sample performance compared to the bootstrap and the delta method. While resembling distributionally robust optimization, we explain the procedural design and develop tight statistical guarantees via a generalization of the empirical likelihood method. The second half develops uncertainty quantification techniques for certifying solution feasibility and optimality in data-driven optimization. Regarding optimality, Chapter 4 proposes a statistical method to estimate the optimality gap of a given solution for stochastic optimization as an assessment of the solution quality. Our approach is based on bootstrap aggregating, or bagging, resampled sample average approximation (SAA). We show how this approach leads to valid statistical confidence bounds for non-smooth optimization. We also demonstrate its statistical efficiency and stability that are especially desirable in limited-data situations. We present our theory that views SAA as a kernel in an infinite-order symmetric statistic. Regarding feasibility, Chapter 5 considers data-driven optimization under uncertain constraints, where solution feasibility is often ensured through a "safe" reformulation of the constraints, such that an obtained solution is guaranteed feasible for the oracle formulation with high confidence. Such approaches generally involve an implicit estimation of the whole feasible set that can scale rapidly with the problem dimension, in turn leading to over-conservative solutions. We investigate validation-based strategies to avoid set estimation by exploiting the intrinsic low dimensionality of the set of all possible solutions output from a given reformulation. We demonstrate how our obtained solutions satisfy statistical feasibility guarantees with light dimension dependence, and how they are asymptotically optimal and thus regarded as the least conservative with respect to the considered reformulation classes.

No description available.

Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.

No description available.

The travel-time flow relationship is not always increasing in nature, it is very difficult to predict precisely. Traditional method fails to replicate this unique conditions. Until millennium, although various researchers and practitioners have given much attention to develop travel-time flow relationships, the advancement to improve travel-time flow relationships was not substantial. The knowledge about the travel-time flow relationship is not commensurate with or parallel to the advancement of new knowledge in other fields. After millennium, most investigators did not devote enough attention to create new knowledge, except for application and performance evaluation of the existing knowledge. Therefore, it is necessary to provide a new theoretical and methodological advancement in travel-time flow relationship. Consequentially, this research proposes a new methodology, which considers stochastic behavior of travel-time flow relationship with probabilistic Bayesian statistics and logistic growth mapping techniques. This research moderately improves the travel-time flow relationship. The unique contribution of this research is that the proposed methods outperforms the existing traditional travel-time flow theory, assumptions, and modeling techniques. The results shows that the proposed model is considerably a good candidate for travel-time predictions. The proposed model performs 36 percent better and accurate travel-time predictions in compared to the existing models. Furthermore, travel-time flow relationship need capacity and free-flow speed estimations. Traditionally, practice of capacity estimation is mostly practical, subjective, and not steady-state capacity. Therefore, a robust and stable capacity-estimation method was developed to eliminate the subjectivity of capacity estimation. The proposed model shows robust and capable of replicating steady-state capacity estimation. The free-flow speed estimation should relate to the traffic-flow speed model while the density is zero. Therefore, this research investigates the existing deterministic speed-density models and recommends a better methodology in free-flow speed estimation. This research presents how the undefined practice of free-flow speed selection can be sensitive. Additionally, finding suitable concurrent travel-time data and traffic volume is crucial and very challenging. To collect concurrent data, this research investigates and develops several technologies such as crowdsource, web app, virtual sensor method, test vehicle, smartphone, global positioning system, and utilized several state and local agencies data collection efforts. Keywords: Travel-Time Flow, Travel-Time Delay, Volume-Delay Function, Travel Time, Origin-Destination Survey, Travel Demand Model, Travel Data Collection, Transportation Survey, Internet Sensor, Crowdsourcing, Virtual Sensor Method, VSM, Transportation Planning, GPS, Smartphone, Loop Detector, Travel -Time Prediction, Travel-Speed Prediction, TDM, Bayesian Inference, Logistic Growth Function.