This thesis describes a PhD project on time series causality analysis and applications. The project is motivated by two EEG measurements of music improvisation experiments, where we aim to use causality measures to construct neural networks to identify the neural differences between improvisation and non-improvisation. The research is based on mathematical backgrounds of time series analysis, information theory and network theory. We first studied a series of popular causality measures, namely, the Granger causality, partial directed coherence (PDC) and directed transfer function (DTF), transfer entropy (TE), conditional mutual information from mixed embedding (MIME) and partial MIME (PMIME), from which we proposed our new measures: the direct transfer entropy (DTE) and the wavelet-based extensions of MIME and PMIME. The new measures improved the properties and applications of their father measures, which were verified by simulations and examples. By comparing the measures we studied, MIME was found to be the most useful causality measure for our EEG analysis. Thus, we used MIME to construct both the intra-brain and cross-brain neural networks for musicians and listeners during the music performances. Neural differences were identified in terms of direction and distribution of neural information flows and activity of the large brain regions. Furthermore, we applied MIME on other EEG and financial data applications, where reasonable causality results were obtained.

This thesis develops new methods to monitor multiple data streams and report some quantity of interest over time. We consider two types of settings. First, we consider a data stream as realisations from a sequence of independent random variables that are revealed over time. To monitor the individual streams, we propose a new type of control chart, based on the cumulative sum chart. Cumulative sum charts are typically used to detect a change in the distribution of a sequence of observations, e.g., shifts in the mean. Usually, after signalling, the chart is restarted by setting it to some value below the signalling threshold. We propose a non-restarting cumulative sum chart which is able to detect periods during which the stream is out of control. Further, we advocate an upper boundary to prevent the cumulative sum chart rising too high, which helps to detect a change back into control. We prove that the non-restarting charts are optimal, in a well-defined sense. Further, we investigate the performance of these charts when the upper boundary is varied. Simulation results show a trade-off between the height of the upper boundary of the chart and the false signal rate. We then present an algorithm to control the false discovery rate across multiple data streams using the non-restarting charts. We consider two definitions of a false discovery: signalling out-of-control when the observations have been in-control since the start and signalling out-of-control when the observations have been in-control since the last time the chart was at zero. We prove that the false discovery rate is controlled under both these definitions simultaneously. Simulations reveal the difference in false discovery rate control when using these and other desirable definitions of a false discovery. In the second setting, a data stream is considered as observations of a Bayesian model revealed over time. The aim is to report a posterior summary of interest quickly and within a user-specified degree of accuracy. A system is presented to tackle such problems. The estimates are calculated using weighted samples stored in a database. The stored samples are maintained such that the accuracy of the estimates and quality of the samples is satisfactory. This maintenance involves varying the number of samples in the database and updating their weights. New samples are generated, when required, by a Markov chain Monte Carlo algorithm. The system is demonstrated using a football league model that is used to predict the end of season table. Correctness of the estimates and their accuracy is shown in a simulation using a linear Gaussian model. Lastly, potential improvements of the system are investigated. A series of motivating simulations illustrate some potential problems of the system. Remedial solutions are suggested, with a view toward implementation in the near future.

The nonlinear stability of two-fluid shear flows in the presence of inertia and/or an insoluble surfactant at the interface is studied in this thesis. Asymptotic analysis in the limit of a thin lower layer is performed and a system of coupled weakly nonlinear evolution equations is derived. The system describes the spatiotemporal evolution of the interface and its local surfactant concentration. It contains a nonlocal term which arises by appropriately matching solutions of the linearised Navier-Stokes equations in the thicker layer to the thin layer solution. The problem corresponding to two-dimensional flows is first solved numerically, by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for asymptotically small and finite Reynolds numbers indicate that the solutions are mostly nonlinear travelling waves or time-periodic waves. As the length of the system increases, the dynamics become more complex and include quasi-periodic and chaotic fluctuations. The stability in three-dimensions of the nonlinear travelling waves observed in two-dimensional flows is also examined. The model derived is also shown to be appropriate in describing interfacial wave structures arising in two-fluid Couette flow experiments. A related two-dimensional dissipative-dispersive partial differential equation is considered in the second part of the thesis. The PDE is similar to the surfactant-free version of the interfacial evolution equation derived in the previous part. A generalisation of that equation with a nonlinearity written in a gradient form provides the well-known two-dimensional Kuramoto-Sivashinsky equation (2D KSE). The 2D KSE has received attention with respect to its mathematical analysis, but numerical solutions are obtained for the first time here. For relatively small domain sizes the solutions are steady states or travelling waves and as the domain becomes increasingly larger, solutions are trapped into a chaotic attractor which is characterised by energy equipartition and symmetry of the energy spectrum.

This thesis develops new tools in stochastic analysis with applications to finance. The first part presents novel developments in the Wiener chaos approach to the modelling, calibration, and pricing of interest rate derivatives. To price financial instruments it suffices to specify the pricing kernel, which in Brownian models can be represented as the conditional variance of a square-integrable random variable which serves as the 'generator' of the pricing kernel. The coefficients of the chaos expansion of the generator act as the parameters of a generic interest-rate model. A special class of generators, arising from 'coherent' chaos expansions, is considered, and the resulting interest rate models are investigated. Coherent representations are important since a kernel generator can be expressed as a linear superposition of coherent generators. This property is exploited to derive general expressions for the pricing kernel, along with the associated discount bond and short rate processes. Pricing formulae for bond options and swaptions are obtained in closed form. The pricing kernel of a generic incoherent model is then obtained by use of the underlying coherent elements. Finite-dimensional representations of coherent chaos models are investigated, and used to construct a class of tractable models having the feature that discount bond prices are piecewise-flat processes. In the second part of the thesis, a general theory of the Wick calculus is developed. Novel results concerning the Wick orders of random variables are derived. In the case where the underlying process is a Brownian motion the Wick calculus reduces to the Ito calculus, but the former is not restricted to the Gaussian class, and is applicable to other cases, such as Lévy processes. With financial applications in mind, the Wick calculus is extended to a wider class of stochastic processes. The thesis concludes with a change of measure analysis for Wick exponentials of Lévy processes, indicating that the Wick calculus can be used as a tool for modelling the dynamics of asset prices.

The viscous drag of a passenger aircraft is influenced significantly by the laminar-turbulent transition in the boundary-layer. Classically the transition process occurs when the turbulence develops as a result of the amplification of instability modes. In the flow past a unswept wing one such mode of instability is observed, Tollmien-Schlichting waves. We deal with the Tollmien-Schlichting waves or, more specifically, with the receptivity of the boundary layer with respect to them. If the surface becomes curved a different instability mode is created: Gortler vortices, which we will investigate as an inviscid secondary instability. Many disturbances can aid the generation of these instabilities but we concentrate on the vibrations of the wing caused by engine noise and the elasticity of the wing itself. This thesis is separated into these two problems of boundary-layer flow over the vibrating wing surface. Firstly we focus on the generation of Tollmien-Schlichting waves due to wing surface vibrations and surface roughness. Piston theory is used to describe the response of the flow outside the boundary layer to wing surface vibrations. Then the perturbations in the Stokes' layer are analysed. The Stokes' layer itself cannot produce a Tollmien-Schlichting wave. Therefore, we will assume that there is a wall roughness. The analysis of the interaction of the Stokes' layer with a wall roughness can be analysed with the help of the Triple deck theory. This allows us to consider the downstream effects of our disturbances and under our flow regime, dependent on the size of wavelength of vibrations, we can Fourier transform and solve our problem for the disturbance pressure. Once we have inverted our solution back into real space with the use of Residue theory we are then able to calculate receptivity coefficients which can be compared to those of previous studies. In the second problem we concentrate on the curved part of the wing where the flow is assumed to be slowly varying in the y direction. This generates Gortler vortices and we expect a sheared base flow with periodicity in the spanwise direction. Using a WKBJ approximation we can derive a multi-scale system of equations, which at leading order can be solved numerically to give eigenvalues and eigenfunctions representing pressure within the boundary-layer. We can only do so with the aid of a two-dimensional problem from which we fix an effective streamwise wavenumber, and an effective maximum growth rate. We use this to create an initial value for our WKBJ eigenvalue. This restricts the values of the streamwise wavenumber that we can calculate solutions for. This also means that when the streamwise wavenumber approaches the effective streamwise wavenumber we get a turning point and hence a breakdown of our WKBJ solutions. We derive and calculate solutions for streamwise wavenumbers away from this limit and discuss the breakdown of these solutions.

An asymptotic scheme is generated that captures the motion of waves within discrete, semi-discrete and continuous periodic media by creating continuum homogenised equations. Conventional homogenisation theory is a well-known classical method valid when the wavelength of any disturbance is long relative to the microstructure. Unfortunately many of the features of interest in real applications involve wave oscillations that are of high frequency and that have wavelength of the same, or similar, order to the microstructure; this requires a new version of homogenisation theory: High frequency homogenisation. This has already been introduced for periodic microstructured continua and extended to discrete systems. Herein we extend high frequency homogenisation further, to deal with localised defect states and non-orthogonal geometries for both discrete and continuous media. We also apply the asymptotic theory to new models, such as in-plane oscillations of the discrete vector system. In each of the studies presented herein, the homogenisation method is verified using numerical and/or analytical solutions.

Multi-relational dynamics are ubiquitous in many complex systems like transportations, social and biological. This thesis studies the two mathematical objects that encapsulate these relationships --- multiplexes and interval graphs. The former is the modern outlook in Network Science to generalize the edges in graphs while the latter was popularized during the 1960s in Graph Theory. Although multiplexes and interval graphs are nearly 50 years apart, their motivations are similar and it is worthwhile to investigate their structural connections and properties. This thesis look into these mathematical objects and presents their connections. For example we will look at the community structures in multiplexes and learn how unstable the detection algorithms are. This can lead researchers to the wrong conclusions. Thus it is important to get formalism precise and this thesis shows that the complexity of interval graphs is an indicator to the precision. However this measure of complexity is a computational hard problem in Graph Theory and in turn we use a heuristic strategy from Network Science to tackle the problem. One of the main contributions of this thesis is the compilation of the disparate literature on these mathematical objects. The novelty of this contribution is in using the statistical tools from population biology to deduce the completeness of this thesis's bibliography. It can also be used as a framework for researchers to quantify the comprehensiveness of their preliminary investigations. From the large body of multiplex research, the thesis focuses on the statistical properties of the projection of multiplexes (the reduction of multi-relational system to a single relationship network). It is important as projection is always used as the baseline for many relevant algorithms and its topology is insightful to understand the dynamics of the system.

This thesis is concerned with studying the hitting time of an absorbing state on Markov chain models that have a countable state space. For many models it is challenging to study the hitting time directly; I present a perturbative approach that allows one to uniformly bound the difference between the hitting time moment generating functions of two Markov chains in a neighbourhood of the origin. I demonstrate how this result can be applied to both discrete and continuous time Markov chains. The motivation for this work came from the field of biology, namely DNA damage and repair. Biophysicists have highlighted that the repair process can lead to Double Strand Breaks; due to the serious nature of such an eventuality it is important to understand the hitting time of this event. There is a phase transition in the model that I consider. In the regime of parameters where the process reaches quasi-stationarity before being absorbed I am able to apply my perturbative technique in order to further understand this hitting time.

The term 'animal spirits' was introduced by Keynes to describe the entrepreneur's often irrational optimism and drive to act as opposed to basing decisions on formal analysis. This PhD thesis provides an analysis, both theoretical and empirical, of this phenomenon in the financial markets from several points of view. In the first chapter we show that the pricing kernel in the economy may be represented in a probabilistic form, as a solution to a stochastic filtering problem. The noise in the associated information process may contain drift term that is impossible to estimate from current market prices of assets. This drift can be associated with 'animal spirits' driving the market. The second chapter is explicitly devoted to 'animal spirits': it introduces a factor based risk-management model for an illiquid project. We show that behavioural factors together with the collateralization mechanism often employed by banks not only increase the risk for the banking system, but also introduce anomalies during high-volatility crisis periods. In the third chapter we apply Hidden Markov Models to estimate animal spirits from historic asset prices. We argue that an arbitrary addition of a stress scenario to the model can greatly improve risk estimation. The last chapter deals with optimal investment problem in a model with behavioural factors. This may be linked to the pricing kernel discussion from the first chapter by the marginal utility maximisation approach to pricing derivatives.

I study a simple model of synchronisation proposed by Jensen (2008). The relevant degrees of freedom are expected to be strictly increasing functions of time, such as the total angle swept out by an oscillator. The model is rooted in Winfree's mean-field model for spontaneous synchronisation; some of Winfree's basic assumptions, such as identical or nearly identical dynamics and identical couplings, are therefore retained. I investigated the behaviour of the present model with respect to synchronisation without and in the presence of time delay. The mathematical treatment focuses on characterising the synchronised state as either attractive or repulsive, producing a theory (which ultimately leads to a phase diagram) that compares well with numerics. I employed a perturbative approach, linearising in small time delays and small phase differences. The interaction between individual oscillators is captured by an interaction matrix, which does not require further approximation, i.e. lattice structure enters exactly. To link with established results in the literature, a mean field theory, however, is also studied. The main result is that these typically systems synchronise due to a time delay.