In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that Q once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that Q makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of Q and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.

Climate modeling and tropical cyclone forecasting are two significant is- sues that are continuously being improved upon for more accurate weather forecasting and preparedness. In this thesis, we have studied three climate models and formulated a new model with a view to determine the outgoing longwave radiation (OLR) budget at the top of the atmosphere (TOA) as ob- served by the National Oceanic and Atmospheric Administration’s (NOAA) satellite based Advanced Very High Resolution Radiometer (AVHRR). In 2006, Karnauskas proposed the African meridional OLR as an Atlantic hur- ricane predictor, the relation was further proven in 2016 by Karnauskas and Li. Here we have considered a similar study for all other tropical cyclone basins.

The ongoing European Refugee Crisis has been one of the most popular trending topics on Twitter for the past 8 months. This paper applies topic modeling on bulks of tweets to discover the hidden patterns within these social media discussions. In particular, we perform topic analysis through solving Non-negative Matrix Factorization (NMF) as an Inexact Alternating Least Squares problem. We accelerate the computation using techniques including tweet sampling and augmented NMF, compare NMF results with different ranks and visualize the outputs through topic representation and frequency plots. We observe that supportive sentiments maintained a strong presence while negative sentiments such as safety concerns have emerged over time.

Topic modeling refers to the process of algorithmically sorting documents into categories based on some common relationship between the documents. This common relationship between the documents is considered the “topic” of the documents. Sentiment analysis refers to the process of algorithmically sorting a document into a positive or negative category depending whether this document expresses a positive or negative opinion on its respective topic. In this paper, I consider the open problem of document classification into a topic category, as well as a sentiment category. This has a direct application to the retail industry where companies may want to scour the web in order to find documents (blogs, Amazon reviews, etc.) which both speak about their product, and give an opinion on their product (positive, negative or neutral). My solution to this problem uses a Non-negative Matrix Factorization (NMF) technique in order to determine the topic classifications of a document set, and further factors the matrix in order to discover the sentiment behind this category of product.

Parameter estimation techniques have been successfully and extensively applied to deterministic models based on ordinary differential equations but are in early development for stochastic models. In this thesis, we first investigate using parameter estimation techniques for a deterministic model to approximate parameters in a corresponding stochastic model. The basis behind this approach lies in the Kurtz limit theorem which implies that for large populations, the realizations of the stochastic model converge to the deterministic model. We show for two example models that this approach often fails to estimate parameters well when the population size is small. We then develop a new method, the MCR method, which is unique to stochastic models and provides significantly better estimates and smaller confidence intervals for parameter values. Initial analysis of the new MCR method indicates that this method might be a viable method for parameter estimation for continuous time Markov chain models.

Blood coagulation is a series of biochemical reactions that take place to form a blood clot. Abnormalities in coagulation, such as under-clotting or over- clotting, can lead to significant blood loss, cardiac arrest, damage to vital organs, or even death. Thus, understanding quantitatively how blood coagulation works is important in informing clinical decisions about treating deficiencies and disorders. Quantifying blood coagulation is possible through mathematical modeling. This review presents different mathematical models that have been developed in the past 30 years to describe the biochemistry, biophysics, and clinical applications of blood coagulation research. This review includes the strengths and limitations of models, as well as suggestions for future work.

This project started early in the summer of 2016 when it became evident there was a need for an effective and efficient signal analysis toolkit for the Littoral Acoustic Demonstration Center Gulf Ecological Monitoring and Modeling (LADC-GEMM) Research Consortium. LADC-GEMM collected underwater acoustic data in the northern Gulf of Mexico during the summer of 2015 using Environmental Acoustic Recording Systems (EARS) buoys. Much of the visualization of data was handled through short scripts and executed through terminal commands, each time requiring the data to be loaded into memory and parameters to be fed through arguments. The vision was to develop a graphical user interface (GUI) that would increase the productivity of manual signal analysis. It has been expanded to make several calculations autonomously for cataloging and meta data storage of whale clicks. Over the last year and a half, a working prototype has been developed with MathWorks matrix laboratory (MATLAB), an integrated development environment (IDE). The prototype is now very modular and can accept new tools relatively quickly when development is completed. The program has been named Banshee, as the mythical creatures are known to “wail”. This paper outlines the functionality of the GUI, explains the benefits of frequency analysis, the physical models that facilitate these analytics, and the mathematics performed to achieve these models.

Bayesian statistics is an important approach to modern statistical analyses. It allows us to use our prior knowledge of the unknown parameters to construct a model for our data set. The foundation of Bayesian analysis is Bayes' Rule, which in its proportional form indicates that the posterior is proportional to the prior times the likelihood. We will demonstrate how we can apply Bayesian statistical techniques to fit a linear regression model and a hierarchical linear regression model to a data set. We will show how to apply different distributions to Bayesian analyses and how the use of a prior affects the model. We will also make a comparison between the Bayesian approach and the traditional frequentist approach to data analyses.

In ecology literature, there is much data which suggests that conspecific negative density dependence (CNDD) and abiotic disturbances increase biodiversity in forests. This thesis elucidates the notion that not only do these two forces increase diversity, but they may also interact with one another in order to achieve higher levels of biodiversity. Abiotic disturbances, like fires and hurricanes, can indirectly impact conspecific effects because when these forces remove individuals from the landscape, the role of the conspecific effects will change. The interaction of these two factors in biodiversity are explored in an agent based forest simulation through a resource surface. Several different types of abiotic disturbances are simulated with either weak or strong CNDD effects in order to establish that different disturbances and conspecific effects cause certain levels of diversity. The underlying causes for the change in impact is also examined.

Material thermal properties are highly sought after to better understand the performance of a material under particular conditions. As new materials are created, their physical properties will determine their performance for various applications. These properties have been estimated using many techniques including experimental testing, numerical modeling, and a combination of both. Existing methods can be time consuming, thus, a time-efficient and precise method to estimate these thermal properties was desired. A one-dimensional finite difference numerical model was developed to replicate the heat transfer through an experimental apparatus. A combination of this numerical model and the Continuous Genetic Algorithm optimization technique was used to estimate material thermal properties of fibrous insulation from test data. The focus of this work was to predict these material thermal properties for an Alumina Paper that is commonly used in aerospace applications. The background, methodology, and results are discussed.