Mathematical Sciences - Doctoral Theses
http://hdl.handle.net/10468/859
Tue, 31 Jan 2023 19:54:23 GMT2023-01-31T19:54:23ZMathematical and computational approaches to contagion dynamics on networks
http://hdl.handle.net/10468/14049
Mathematical and computational approaches to contagion dynamics on networks
Humphries, Rory
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.
Tue, 27 Sep 2022 00:00:00 GMThttp://hdl.handle.net/10468/140492022-09-27T00:00:00ZOption pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method
http://hdl.handle.net/10468/13565
Option pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method
Trinh, Yen Thuan
The 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.
Mon, 18 Jul 2022 00:00:00 GMThttp://hdl.handle.net/10468/135652022-07-18T00:00:00ZThe role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model
http://hdl.handle.net/10468/13582
The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model
Maulana, Heru
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.
Mon, 04 Jul 2022 00:00:00 GMThttp://hdl.handle.net/10468/135822022-07-04T00:00:00ZAlgebraic central limit theorems in noncommutative probability
http://hdl.handle.net/10468/12591
Algebraic central limit theorems in noncommutative probability
Alahmade, Ayman
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.
Mon, 24 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10468/125912022-01-24T00:00:00Z