Mathematical Sciences - Doctoral Theses
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- ItemMathematical and computational approaches to contagion dynamics on networks(University College Cork, 2022-09-27) Humphries, Rory; Hoevel, Philipp; Mulchrone, Kieran F.In this thesis, we firstly introduce the basic terminology and concepts needed for the the following chapters. In particular we introduce the basics of graph/network theory, epidemiological models (both well mixed and on networks), and mobility models (the gravity and radiation models). After the introduction of these topics, we propose a general framework for epidemiological network models from which the known individual-based and pair-based models can be derived. We then introduce a more exact pair-based model by showing previous iterations are a linearised version of it, and then we extend it further to the temporal setting. Next, we present a meta-population model for the spread of COVID-19 in Ireland which makes use of temporal commuting patters generated from the radiation model. Finally, we analyse a year worth of Irish cattle trade data. We then fit a number of mobility models and show that an altered version of the radiation model, which we call the generalised radiation model, is able to accurately reproduce the distance distribution of cattle trades in the country.
- ItemThe role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model(University College Cork, 2022-07-04) Maulana, Heru; Kelly, Conall; Indonesia Endowment Fund for Education (LPDP)We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller’s condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller’s condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller’s condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.
- ItemOption pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method(University College Cork, 2022-07-18) Trinh, Yen Thuan; Hanzon, Bernard; Ma, JingtangThe thesis introduces a new method, the MC-Tree method, for pricing certain financial derivatives, especially options with high accuracy and efficiency. Our solution is to combine Monte Carlo (MC) method and Tree method by doing a mixing distribution on the tree, and the output is the compound distribution on the tree. The compound distribution in the tree output (after a logarithmic transformation of the asset prices) is not the ideal Gaussian distribution but has entropy values close to the maximum possible Gaussian entropy. We can get closer using entropy maximization. We introduce two correction techniques: distribution correction and bias correction to improve the accuracy and completeness of the model. The thesis presents an algorithm and numerical results for calculations of CVA on an American put option using the MC-Tree method. The MC-Tree method with the distribution correction technique significantly improves accuracy, resulting in practically exact solutions, compared to analytical solutions, at the tree depth $N=50$ or $100$ and MC-drawings $M=10^5$. The bias-correction technique makes the resulting tree model complete in the sense of financial mathematics and obtains the risk-neutral probability. Besides, we have obtained new formulae for the calculations of the entropy and the Kullback-Leibler divergence for rational densities and approximate entropy of finite Gaussian mixture.
- ItemApplication of cointegrated state-space models to financial and economic data(University College Cork, 2021-02-01) Al-qurashi, Miaad Hamad; Hanzon, BernardIn 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.
- ItemAlgebraic central limit theorems in noncommutative probability(University College Cork, 2022-01-24) Alahmade, Ayman; Koestler, Claus; Taibah UniversityDistributional 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.