In this thesis, we study two classes of point processes by analysing key properties and discussing applications in finance and insurance. The first point process studied was the default indicator process in credit risk modelling. We considered a pure jump Lévy process of finite variation for the asset value and an unobservable random barrier. The default time was defined as the first time the asset value falls below the barrier. Using the indistinguishable intensity process and the instantaneous likelihood process, we proved the absolute continuity of the compensator for the default indicator process, or equivalently, the existence of the intensity process of the default time. Moreover, we found the explicit representation of the intensity in terms of the distance between the asset value and its running minimal value, thus the intensity is an endogenous process, which sheds new light on the relationship between the intensity model and the structural model. The second class of point processes is the Dynamic Contagion Process, which has intensities modelled with a shot-noise component describing the external impact and mutually-exciting jump components that describe the internal contagion effect. In the bivariate case, we found the stationarity condition with which we explored the diffusion approximation of the high frequency point process system and applied it in filtering. In the univariate case, we constructed a pure jump process derived from a dynamic contagion process and showed the weak convergence to a Cox-Ingersoll- Ross model (CIR) process. The pathwise approximation provides an alternative method of simulating the square-root processes and can be further extended to the approximation of the Heston model in option pricing.

This thesis presents analysis and computations of systems of nonlinear partial differential equations (PDEs) modelling the dynamics of three stratified immiscible viscous layers flowing inside a channel with parallel walls inclined to the horizontal. The three layers are separated by two fluid-fluid interfaces that are free to evolve spatiotemporally and nonlinearly when the flow becomes unstable. The determination of the flow involves solution of the Navier-Stokes in domains that are changing due to the evolution of the interfaces whose position must be determined as part of the solution, providing a hard nonlinear moving boundary problem. Long-wave approximation and a weakly nonlinear analysis of the Navier-stokes equations along with the associated boundary conditions, leads to reduced systems of nonlinear PDEs that in general form are systems of coupled Kuramoto- Sivashinsky equations. These physically derived coupled systems are mathematically rich due to the rather generic presence of coupled nonlinearities that undergo hyperbolic-elliptic transitions, along with high order dissipation. Analysis and numerical computations of the resulting coupled PDEs is presented in order to understand the stability of multilayer channel flows and explore and quantify the different types of underlying nonlinear phenomena that are crucial in applications. Importantly, it is found that multilayer flows can be unstable even at zero Reynolds numbers, in contrast to single interface problems. Furthermore, the thesis investigates the dynamical behaviour of the zero viscosity limits of the derived systems in order to verify their physical relevance as reduced models. Strong evidence of the existence of the zero viscosity limit is provided for mixed hyperbolic-elliptic type systems whose global existence is an open and challenging mathematical problem. Finally, a novel sufficient condition is derived for the occurrence of hyperbolic-elliptic transitions in general conservation laws of mixed type; the condition is demonstrated for several physical systems that have been studied in the literature.

The objects of study in this thesis are Hamiltonian systems of ordinary differential equations possessing homoclinic orbits. A homoclinic orbit is a solution of the system which converges to the same invariant set as time approaches both positive and negative infinity. In our case, the invariant set in question is assumed to be a nonhyperbolic equilibrium state. Such an equilibrium state possesses a center manifold, containing all orbits which remain close to the equilibrium. We are concerned with finding orbits which converge to orbits in the center manifold in both time directions. We consider firstly the case in which the nonhyperbolic eigenvalues at the equilibrium consist of pairs of nonzero purely imaginary eigenvalues. We study the set of homoclinics to the center manifold by constructing an operator on a suitable function space whose zeros correspond to homoclinics. We use a Lyapunov-Schmidt technique to reduce the problem to that of studying the zero set of a real-valued function defined on the center manifold, which has a critical point at the origin. A formula is found for the Hessian matrix at this critical point, involving the so called scattering matrix. Under nonresonance and nondegeneracy conditions, we characterise the possible Morse indices of the Hessian, permitting an application of the Morse lemma to describe the set of homoclinics. We also consider special cases, including reversible systems. We then consider a more geometric approach to the problem, allowing us to define a nonlinear analogue of the scattering matrix using stable and unstable foliations of the invariant manifolds. We use this approach to unfold the system in parametrised families - we consider here also the case of a two dimensional center manifold corresponding to zero eigenvalues - bifurcation diagrams are produced for homoclinics to the origin in this case. The effects of additional reversible structure are again considered.

Modular forms play a central and critical role in the study of modern number theory. These remarkable and beautiful functions have led to many spectacular results including, most famously, the proof of Fermat's Last Theorem. In this thesis we find connections between these enigmatic objects. In particular, we describe the construction and properties of a singular theta lift, closely related to the well known Shimura correspondence. We first define a (twisted) lift of harmonic weak Maass forms of weight 3/2-k, by integrating against a well chosen kernel Siegel theta function. Using this, we obtain a new class of automorphic objects in the upper-half plane of weight 2-2k for the group Gamma_{0}(N). We reveal these objects have intriguing singularities along a collection of geodesics. These singularities divide the upper-half plane into Weyl chambers with associated wall crossing formulas. We show our lift is harmonic away from the singularities and so is an example of a locally harmonic Maass form. We also find an explicit Fourier expansion. The Shimura/Shintani lifts provided very important correspondences between half-integral and even weight modular forms. Using a natural differential operator we link our lift to these. This connection then allows us to derive the properties of the Shimura lift. The nature of the singularities suggests we formulate all of these ideas as distributions and finally we consider the current equation encompassing them. This work provides extensions of the theta lifts considered by Borcherds (1998), Bruinier (2002), Bruinier and Funke (2004), Hövel (2012) and Bringmann, Kane and Viazovska (2013).

In this thesis, we study time-inhomogeneous and McKean-Vlasov type stochastic differential equations (SDEs), along with related partial differential equations (PDEs). We are particularly interested in regularity estimates and their applications to numerical methods. In the first part of the thesis, we build on the work of Kusuoka \& Stroock to develop sharp estimates on the derivatives of solutions to time-inhomogeneous parabolic PDEs. The basis of these estimates is an integration by parts formula for derivatives of the solution under the UFG condition, which is weaker than the uniform Hoermander condition. This integration by parts formula is obtained using Malliavin Calculus. The formula allows us to extend the notion of classical solution to a framework where differentiability does not necessarily hold in all directions. As an application, we extend the error analysis for the cubature on Wiener space method to time-inhomogeneous stochastic differential equations. We then present two cubature on Wiener space algorithms for the numerical solution of McKean-Vlasov SDEs with smooth scalar interaction. The analysis involves the regularity estimates proved previously and takes place under a uniform strong Hoermander condition. Finally, we develop integration by parts formulas on Wiener space for solutions of SDEs with general McKean-Vlasov interaction and uniformly elliptic coefficients. These formulas hold both for derivatives with respect to a real variable and derivatives with respect to a measure in the sense of Lions. This allows us to develop estimates on the density of solutions of the McKean-Vlasov SDEs. We also prove the existence of a classical solution to a related PDE with irregular terminal condition.

The main objective of this thesis is to propose approximations to option sensitivities in stochastic volatility models. The first part explores sequential Monte Carlo techniques for approximating the latent state in a Hidden Markov Model. These techniques are applied to the computation of Greeks by adapting the likelihood ratio method. Convergence of the Greek estimates is proved and tracking of option prices is performed in a stochastic volatility model. The second part defines a class of approximate Greek weights and provides high-order approximations and justification for extrapolation techniques. Under certain regularity assumptions on the value function of the problem, Greek approximations are proved for a fully implementable Monte Carlo framework, using weak Taylor discretisation schemes. The variance and bias are studied for the Delta and Gamma, when using such discrete-time approximations. The final part of the thesis introduces a modified explicit Euler scheme for stochastic differential equations with non-Lipschitz continuous drift or diffusion; a strong rate of convergence is proved. The literature on discretisation techniques for stochastic differential equations has been motivational for the development of techniques preserving the explicitness of the algorithm. Stochastic differential equations in the mathematical finance literature, including the Cox-Ingersoll-Ross, the 3/2 and the Ait-Sahalia models can be discretised, with a strong rate of convergence proved, which is a requirement for multilevel Monte Carlo techniques.

We investigate the linear instability and nonlinear stability for some convection models, and present results and details of their computation in each case. The convection models we consider are: convection in a variable gravity field with magnetic field effect; magnetic effect on instability and nonlinear stability in a reacting fluid; magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid; Poiseuille flow in a porous medium with slip boundary conditions. The structural stability for these convection models is studied. A priori bounds are derived. With the aid of these a priori bounds we are able to demonstrate continuous dependence of solutions on some coefficients. We further show that the solution depends continuously on a change in the coefficients. Chebyshev collection, finite element, finite difference, high order finite difference methods are also developed for the evaluation of eigenvalues and eigenfunctions inherent in stability analysis in fluid and porous media, drawing on the experience of the implementation of the well established techniques in the previous work. These generate sparse matrices, where the standard homogeneous boundary conditions for both porous and fluid media problems are contained within the method. When the difference between the linear (which predicts instability) and nonlinear (which predicts stability) thresholds is very large, the validity of the linear instability threshold to capture the onset of the instability is unclear. Thus, we develop a three dimensional simulation to test the validity of these thresholds. To achieve this we transform the problem into a velocity-vorticity formulation and utilise second order finite difference schemes. We use both implicit and explicit schemes to enforce the free divergence equation.

This thesis discusses periodic one dimensional arrays of BPS monopoles. An approximation based on the spectral curve is shown to provide an increasingly accurate description of the monopole fields in the limit of large monopole size to period ratio. Away from this limit the periodic monopole is studied by means of the Nahm transform, which leads to a dual system of Hitchin equations on a cylinder. A combination of analytical and numerical techniques is used to study the spatial symmetries of the periodic 2-monopole and its moduli space. In particular, the asymptotic moduli space metric is determined from the Nahm data, and symmetric one parameter families of monopole scattering processes are identified through the core of the moduli space. These ideas are readily applicable to higher charge periodic monopoles.

In this thesis I explore the connection between geometry and quantum entanglement, in the context of holographic duality, where entanglement entropies in a quantum field theory are associated with the areas of surfaces in a dual gravitational theory. The first chapter looks at a phase transition in such systems in finite size and at finite temperature, associated with the properties of minimal surfaces in a static black hole background. This is followed by the related problem of extremal surfaces in a spacetime describing the dynamical process of black hole formation, with a view towards understanding the connections between bulk locality and various field theory observables including entanglement entropy. The third chapter looks at the simple case of pure gravity in three spacetime dimensions, where I show how evaluating the entanglement entropy can be reduced to a simple algebraic calculation, and apply it to some interesting examples. Finally, the role played by topology of surfaces in a proposed derivation of a holographic entanglement entropy formula is investigated. This makes it clear what assumptions are required in order to reproduce the ‘homology constraint’, a topological condition necessary for consistency with field theory.

The presented thesis contains research on topological solitons in (2+1) and (3+1) dimensional classical field theories, focusing upon the Skyrme model. Due to the highly non-linear nature of this model, we must consider various numerical methods to find solutions. We initially consider the (2+1) baby Skyrme model, demonstrating that the currently accepted form of minimal energy solutions, namely straight chains of alternating phase solitons, does not hold for higher charge. Ring solutions with relative phases changing by pi for even configurations or pi-pi/B for odd numbered configurations, are demonstrated to have lower energy than the traditional chain configurations above a certain charge threshold, which is dependant on the parameters of the model. Crystal chunk solutions are then demonstrated to take a lower energy but for extremely high values of charge. We also demonstrate the infinite charge limit of each of the above configurations. Finally, a further possibility of finding lower energy solutions is discussed in the form of soliton networks involving rings/chains and junctions. The dynamics of some of these higher charge solutions are also considered. In chapter 3 we numerically simulate the formation of (2+1)-dimensional baby Skyrmions from domain wall collisions. It is demonstrated that Skyrmion, anti-Skyrmion pairs can be produced from the interaction of two domain walls, however the process can require quite precise conditions. An alternative, more stable, formation process is proposed and simulated as the interaction of more than two segments of domain wall. Finally domain wall networks are considered, demonstrating how Skyrmions may be produced in a complex dynamical system. The broken planar Skyrme model, presented in chapter 4, is a theory that breaks global O(3) symmetry to the dihedral group D_N. This gives a single soliton solution formed of N constituent parts, named partons, that are topologically confined. We show that the configuration of the local energy solutions take the form of polyform structures (planar figures formed by regular N-gons joined along their edges, of which polyiamonds are the N=3 subset). Furthermore, we numerically simulate the dynamics of this model. We then consider the (3+1) SU(2) Skyrme model, introducing the familiar concepts of the model in chapter 5 and then numerically simulating their formation from domain walls. In analogue with the planar case, it is demonstrated that the process can require quite precise conditions and an alternative, more stable, formation process can be achieved with more domain walls, requiring far less constraints on the initial conditions used. The results in chapter 7 discuss the extension of the broken baby Skyrme model to the 3-dimensional SU(2) case. We first consider the affect of breaking the isospin symmetry by altering the tree level mass of one of the pion fields breaking the SO(3) isospin symmetry to an SO(2) symmetry. This serves to exemplify the constituent make up of the Skyrme model from ring like solutions. These rings then link together to form higher charge solutions. Finally the mass term is altered to allow all the fields to have an equivalent tree level mass, but the symmetry of the Lagrangian to be broken, firstly to a dihedral symmetry D_N and then to some polyhedral symmetries. We now move on to discussing both the baby and full SU(2) Skyrme models in curved spaces. In chapter 8 we investigate SU(2) Skyrmions in hyperbolic space. We first demonstrate the link between increasing curvature and the accuracy of the rational map approximation to the minimal energy static solutions. We investigate the link between Skyrmions with massive pions in Euclidean space and the massless case in hyperbolic space, by relating curvature to the pion mass. Crystal chunks are found to be the minimal energy solution for increased curvature as well as increased mass of the model. The dynamics of the hyperbolic model are also simulated, with the similarities and differences to the Euclidean model noted. One of the difficulties of studying the full Skyrme model in (3+1) dimensions is a possible crystal lattice. We hence reduce the dimension of the model and first consider crystal lattices in (2+1)-dimensions. In chapter 9 we first show that the minimal energy solutions take the same form as those from the flat space model. We then present a method of tessellating the Poincare disc model of hyperbolic space with a fundamental cell. The affect this may have on a resulting Skyrme crystal is then discussed and likely problems in simulating this process. We then consider the affects of a pure AdS background on the Skyrme model, starting with the massless baby Skyrme model in chapter 10. The asymptotics and scale of charge 1 massless radial solutions are demonstrated to take a similar form to those of the massive flat space model, with the AdS curvature playing a similar role to the flat space pion mass. Higher charge solutions are then demonstrated to exhibit a concentric ring-like structure, along with transitions (dubbed popcorn transitions in analogy with models of holographic QCD) between different numbers of layers. The 1st popcorn transitions from an n layer to an n+1-layer configuration are observed at topological charges 9 and 27 and further popcorn transitions for higher charges are predicted. Finally, a point-particle approximation for the model is derived and used to successfully predict the ring structures and popcorn transitions for higher charge solitons. The final chapter considers extending the results from the penultimate chapter to the full SU(2) model in a pure AdS_4 background. We make the prediction that the multi-layered concentric ring solutions for the 2-dimensional case would correlate a multi-layered concentric rational map configuration for the 3-dimensional model. The rational map approximation is extended to consider multi-layered maps and the energies demonstrated to reduce the minimal energy solution for charge B=11 which is again dubbed a popcorn transition. Finally we demonstrate that the multi shell structure extends to the full field solutions which are found numerically. We also discuss the affect of combined symmetries on the results which (while likely to be important) appear to be secondary to the dominant effective potential of the metric which simulates a packing problem and hence forces the popcorn transitions to act accordingly with the 2-dimensional model.