Mathematical Sciences - Doctoral Theseshttps://hdl.handle.net/10468/8592024-10-07T07:55:18Z2024-10-07T07:55:18Z361A study of the Hobson and Rogers volatility modelRyan, Gearóidhttps://hdl.handle.net/10468/20362023-04-04T07:36:13Z2014-01-01T00:00:00Zdc.title: A study of the Hobson and Rogers volatility model
dc.contributor.author: Ryan, Gearóid
dc.description.abstract: We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.
2014-01-01T00:00:00ZAlgebraic central limit theorems in noncommutative probabilityAlahmade, Aymanhttps://hdl.handle.net/10468/125912023-04-04T10:44:00Z2022-01-24T00:00:00Zdc.title: Algebraic central limit theorems in noncommutative probability
dc.contributor.author: Alahmade, Ayman
dc.description.abstract: Distributional symmetries and invariance principles in noncommutative probability theory provide sufficient conditions for the existence of central limit laws. In contrast to classical probability theory, there exist many different central limit laws for exchangeable sequences of noncommutative random variables and still little is known about their concrete form. This thesis goes one step further and investigates central limit laws for non-exchangeable spreadable sequences in the context of *-algebraic probability spaces. This provides first results on a new type of combinatorics underlying multivariate central limit theorems (CLTs).
The starting point of the thesis has been a quite simple family of spreadable sequences, which is parametrized by a unimodular complex parameter ω. Each sequence of this family is spreadable, but not exchangeable for ω different from ±1. Moreover, the sequences from this family provide CLTs, which interpolate between the normal distribution (ω = 1) and the symmetric Bernoulli distribution (ω = −1), but differ from q-Gaussian distributions (−1 < q < 1). An algebraic structure, which underlies the considered family, is identified and used to define so-called ‘ω-sequences of partial isometries’. These ω-sequences encode all information, as it is relevant for computations of *-algebraic CLTs. Explicit combinatorial formulas are established for CLTs associated to such ω-sequences, which involve the counting of oriented crossings of directed ordered pair partitions. The limiting distributions of certain multivariate CLTs associated to ω-sequences show some features as they are defining for ‘z-circular systems’ in the work of Mingo and Nica. This similarity, as well as the well-known relation between q-circular systems and q-semicircular systems (for −1 ≤ q ≤ 1), guides the introduction of ‘z-semicircular systems’ in this thesis. Finally, it is shown that the class of z-semicircular systems is stable under certain multivariate central limits. In other words, the moment formulas of z-semicircular systems are reproduced in large N-limit formulas of central limit type.
2022-01-24T00:00:00ZAn exploration of open coupled-cavity lasers: from exceptional points to dynamicsO'Connor, Christopher P.J.https://hdl.handle.net/10468/118862023-04-04T11:03:53Z2020-12-24T00:00:00Zdc.title: An exploration of open coupled-cavity lasers: from exceptional points to dynamics
dc.contributor.author: O'Connor, Christopher P.J.
dc.description.abstract: In optical communications, there is an increase in demand for investigation of multisection laser cavities, which are involved in the development of Photonic Integrated Circuits or PICs. The requirement for such circuits results from, for example, the increase in demand for faster internet devices, which means improving the infrastructure involved. PICs are useful for such advancements as instead of an electrical circuit, these use optical devices such as lasers. Furthermore, these laser cavities are closely interacting or strongly coupled as they are of millimetre to submillimetre scales. Due to the size and complexity of these devices, they are not yet fully understood. Previous work has shown some interesting behaviour with these strongly coupled cavities. For example, the existence of exceptional points, where two modes coalesce, and also the possibility for a laser mode to go below threshold with increasing population inversion in a cavity, all in the steady state. However, for these types of devices, there is currently no full dynamical model that considers the complexity of strongly coupled cavities, with only outgoing light at the boundary of the laser.
Thus, the aim of this thesis is to investigate three important areas. The first is to understand the steady state situation further and improve upon what is known already. We introduce a new basis to do this and create a threshold condition, which is used to explore the complexities of coupled cavities. With this, we further address the occurrence of these exceptional points and give a more in-depth insight into the interesting effects that occur. The second area is to introduce a new, elegant approach in solving the electromagnetic field equation for the steady state, while still considering the geometrical features of a multisection laser. We introduce a second formalism for laser equations at threshold, which is shown to be a projection of a loxodromic spiral on a Riemann sphere with the use of Mobius transformations. With this, we investigate the threshold branches of multisection laser devices and discover a different type of exceptional point, where instead, the branches merge rather than modes. Furthermore, this new approach removes the need for unnecessary information while retaining the important physical characteristics to explain a coupled cavity laser. Lastly, a dynamical model that represents a strongly coupled laser with outgoing boundary conditions is introduced by connecting the classical electromagnetic field to the quantum mechanical description for the active medium. We go beyond the steady state by introducing a time-dependent scaling term for the electromagnetic field, which scales with the population inversion equations to provide a physically explained self-consistent set of dynamical equations. This model confirms results seen in the steady state situation. It also expands upon the steady state to show possible dynamical properties which are a result of the close interactions between both cavities, while crucially considering open boundary conditions while using the spatial profile for the active medium.
2020-12-24T00:00:00ZAn investigation of post-primary students' images of mathematicsLane, Ciara Mary Franceshttps://hdl.handle.net/10468/10962023-04-04T07:00:51Z2013-01-01T00:00:00Zdc.title: An investigation of post-primary students' images of mathematics
dc.contributor.author: Lane, Ciara Mary Frances
dc.description.abstract: This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.
2013-01-01T00:00:00ZApplication of cointegrated state-space models to financial and economic dataAl-qurashi, Miaad Hamadhttps://hdl.handle.net/10468/131852023-04-04T10:54:21Z2021-02-01T00:00:00Zdc.title: Application of cointegrated state-space models to financial and economic data
dc.contributor.author: Al-qurashi, Miaad Hamad
dc.description.abstract: In the dynamic stochastic modeling of financial and economic time series, the concept of cointegration plays an important role. It refers to the existence of long-term equilibrium relations between variables in a dynamic environment. The cointegration theory posits that, in a non-stationary environment, the long-term equilibrium relations will show up as stationary relations between certain variables. In the context of linear models, this translates into the existence of so-called cointegrating linear relations and corresponding cointegrating vectors.
The error correction model was introduced by Engle and Granger(1987) for estimating models exhibiting cointegration. However, it was not until Johansen(1988, 1991) that a rigorous methodology was proposed for estimation and analysis of cointegrated models in a multivariate dynamic setting. The iconic research by Johansen sets out the Vector Error Correction Model (VECM) approach to cointegration, by applying maximum likelihood estimation. The VECM is based on the Vector Autoregressive (VAR) model. The VAR model has several shortcomings for cointegration studies. Several authors have explored estimation of cointegration using VARMA and state-space models, e.g. Ribarits and Hanzon (2014b,a), Yap and Reinsel (1995), Lütkepohl and Claessen (1997), Poskitt (1994, 2006), Bauer and Wagner (2002), Kascha and Trenkler (2011), among others.
This research study focuses on the discrete-time state-space model as well as on a continuous-discrete time state-space model. This research study proposes a parameterization and an estimation method that translates the cointegration property into a low-rank constraint on a resulting likelihood optimization. First, the study partially optimizes the likelihood function. The resulting criterion is a function of eigenvalues of a matrix due to the low-rank constraint. This raises the challenge of calculating the derivatives of the parameterized matrix eigenvalues. The research study addresses this problem by applying the envelope theorem. The study applies a gradient method to optimize the remaining model parameters using a new parametrization. This parameterization has bounded and numerically stable parameters. The method is illustrated by some simulated examples and examples involving stock price index data, oil price data, and more. Comparisons are made against a classical VECM, and in many cases, the state-space model yields better results.
2021-02-01T00:00:00ZCluster sampling in large oral health surveys – issues and implications for design and analysisSheerin, Anthony Josephhttps://hdl.handle.net/10468/101092023-04-04T11:04:21Z2019-10-09T00:00:00Zdc.title: Cluster sampling in large oral health surveys – issues and implications for design and analysis
dc.contributor.author: Sheerin, Anthony Joseph
dc.description.abstract: The use of survey sampling has become routine in almost all aspects of life. If the chosen sample is representative of the population, inferences about the population can be made from the sample. Chapter 1 reviews commonly used survey sampling techniques and discusses the variance estimation methods required for these techniques. Particular attention will be given to cluster sampling, as the data underlying this thesis was gathered in such a manner. The results of the literature review in Chapter 1 identify the direction of analysis for Chapter 2. Chapter 2 compares the readily available cluster variance estimation methods, namely Taylor Series Linearisation (TSL) and the delete-one jackknife (JK1), on simulated finite populations of data with known characteristics, to ascertain if there are any situations where the estimates differ. Multiple situations are examined systematically, including skewed distributions, large sampling fractions and small sample sizes with these situations occurring both in isolation and simultaneously. These methods provided identical estimates when the sampling fraction was small, and the number of observations, particularly at second-stage, was large. However, when these conditions were violated, diverging estimates occurred. Moreover, design effects less than one are seen in this chapter which is unusual in a cluster sampling setting. Chapter 3 looks at using the above variance estimation techniques on a national oral health dataset. Chapter 3 analyses a national oral health dataset, in which there are regions with design effects less than one. As the data was collected using cluster sampling, the presence of design effects less than one is extremely unusual and warranted an analysis to try and identify the possible causes. Both cluster analysis and linear model methods are used to identify variables which may indicate the presence of a design effect less than one, with the number of clusters sampled (n) being the most identified variable using different models. Chapter 3 suggests that reducing the number of clusters sampled will reduce the design effect of the data. Chapter 4 looks at the effect of reducing the number of clusters sampled (n) on the SE and DE estimates of the survey data analysed in Chapter 3. Variance estimates are produced using a jackknife resampling approach, looking at all combinations when n = 1,2,…,10 clusters are dropped in turn. The results show that a small number of clusters per community care area (CCA) can be dropped, with no impact on the bias of the estimate and a very small increase in the SE of the estimate.
2019-10-09T00:00:00ZConstant mean curvature surfaces and Heun's differential equationsMota, Eduardohttps://hdl.handle.net/10468/83752023-04-04T07:21:23Z2019-01-01T00:00:00Zdc.title: Constant mean curvature surfaces and Heun's differential equations
dc.contributor.author: Mota, Eduardo
dc.description.abstract: This thesis is concerned with the problem of constructing surfaces of constant mean curvature with irregular ends by using the class of Heun’s Differential Equations. More specifically, we are interested in obtaining immersion of punctured Riemann spheres into three dimensional Euclidean space with constant mean curvature. These immersions can be described by a Weierstrass representation in terms of holomorphic loop Lie algebra valued 1-forms. We describe how to encode each of the differential equations in Heun’s family in the Weierstrass representation. Next, we investigate monodromy problems for each of the cases in order to ensure periodicity of all the resulting immersions. This allows us to find four families of surfaces with constant mean curvature and irregular ends. These families can be described as trinoids, cylinders, perturbed Delaunay surfaces and planes. Finally, we study some symmetry properties of these groups of surfaces.
2019-01-01T00:00:00ZDifferential and numerical models of hysteretic systems with stochastic and deterministic inputsMcCarthy, Stephen P.https://hdl.handle.net/10468/11382023-04-04T07:09:08Z2013-01-01T00:00:00Zdc.title: Differential and numerical models of hysteretic systems with stochastic and deterministic inputs
dc.contributor.author: McCarthy, Stephen P.
dc.description.abstract: Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.
2013-01-01T00:00:00ZFinancial modelling with 2-EPT probability density functionsSexton, Hugh Conorhttps://hdl.handle.net/10468/14302023-04-04T07:40:26Z2013-01-01T00:00:00Zdc.title: Financial modelling with 2-EPT probability density functions
dc.contributor.author: Sexton, Hugh Conor
dc.description.abstract: The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.
2013-01-01T00:00:00ZFlat surfaces of finite type in the 3-sphereMcCarthy, Alanhttps://hdl.handle.net/10468/19352023-04-04T07:09:28Z2014-01-01T00:00:00Zdc.title: Flat surfaces of finite type in the 3-sphere
dc.contributor.author: McCarthy, Alan
dc.description.abstract: We introduce the notion of flat surfaces of finite type in the 3- sphere, give the algebro-geometric description in terms of spectral curves and polynomial Killing fields, and show that finite type flat surfaces generated by curves on S2 with periodic curvature functions are dense in the space of all flat surfaces generated by curves on S2 with periodic curvature functions.
2014-01-01T00:00:00Z