Binocular rivalry is an interesting phenomenon where perception oscillates between different images presented to the two eyes. This thesis is primarily concerned with modelling travelling waves of visual perception during transitions between these perceptual states. In order to model this effect in such a way that we retain as much analytical insight into the mechanisms as possible we employed neural field theory. That is, rather than modelling individual neurons in a neural network we treat the cortical surface as a continuous medium and establish integro-differential equations for the activity of a neural population. Our basic model which has been used by many previous authors both within and outside of neural field theory is to consider a one dimensional network of neurons for each eye. It is assumed that each network responds maximally to a particular feature of the underlying image, such as orientation. Recurrent connections within each network are taken to be excitatory and connections between the networks are taken to be inhibitory. In order for such a topology to exhibit the oscillations found in binocular rivalry there needs to be some form of slow adaptation which weakens the cross-connections under continued firing. By first considering a deterministic version of this model, we will show that, in fact, this slow adaptation also serves as a necessary "symmetry breaking" mechanism. Using this knowledge to make some mild assumptions we are then able to derive an expression for the shape of a travelling wave and its wave speed. We then go on to show that these predictions of our model are consistent not only with numerical simulations but also experimental evidence. It will turn out that it is not acceptable to completely ignore noise as it is a fundamental part of the underlying biology. Since methods for analyzing stochastic neural fields did not exist before our work, we first adapt methods originally intended for reaction-diffusion PDE systems to a stochastic version of a simple neural field equation. By regarding the motion of a stochastic travelling wave as being made up of two distinct components, firstly, the drift-diffusion of its overall position, secondly, fast fluctuations in its shape around some average front shape, we are able to derive a stochastic differential equation for the front position with respect to time. It is found that the front position undergoes a drift-diffusion process with constant coefficients. We then go on to show that our analysis agrees with numerical simulation. The original problem of stochastic binocular rivalry is then re-visited with this new toolkit and we are able to predict that the first passage time of a perceptual wave hitting a fixed barrier should be an inverse Gaussian distribution, a result which could potentially be experimentally tested. We also consider the implications of our stochastic work on different types of neural field equation to those used for modelling binocular rivalry. In particular, for neural fields which support pulled fronts propagating into an unstable state, the stochastic version of such an equation has wave fronts which undergo subdiffusive motion as opposed to the standard diffusion in the binocular rivalry case.

The work in this thesis focuses on the development of a Stochastic Model Predictive Control (SMPC) algorithm for linear systems with additive and multiplicative stochastic uncertainty subjected to linear input/state constraints. Constraints can be in the form of hard constraints, which must be satisfied at all times, or soft constraints, which can be violated up to a pre-defined limit on the frequency of violation or the expected number of violations in a given period. When constraints are included in the SMPC algorithm, the difficulty arising from stochastic model parameters manifests itself in the online optimization in two ways. Namely, the difficulty lies in predicting the probability distribution of future states and imposing constraints on closed loop responses through constraints on predictions. This problem is overcome through the introduction of layered tubes around a centre trajectory. These tubes are optimized online in order to produce a systematic and less conservative approach of handling constraints. The layered tubes centered around a nominal trajectory achieve soft constraint satisfaction through the imposition of constraints on the probabilities of one-step-ahead transition of the predicted state between the layered tubes and constraints on the probability of one-step-ahead constraint violations. An application in the field of Sustainable Development policy is used as an example. With some adaptation, the algorithm is extended the case where the uncertainty is not identically and independently distributed. Also, by including linearization errors, it is extended to non-linear systems with additive uncertainty.

To prevent loss of lives during seasonal disasters, relief agencies distribute critical supplies and provide lifesaving services to the affected populations. Despite agencies' efforts, frequently occuring disasters increase the cost of relief operations. The purpose of our study is to minimize the cost of relief operations, considering that such disasters cause random demand. To achieve this, we have formulated a series of models, which are distinct from the current studies in three ways. First, to the best of our knowledge, we are the first ones to capture both perishable and durable products together. Second, we have aggregated multiple products in a different way than current studies do. This unique aggregation requires less data than that of other types of aggregation. Finally, our models are compatible with the practical data generated by FEMA. Our models offer insights on the impacts of various parameters on optimum cost and order size. The analyses of correlation of demand and quality of information offer interesting insights; for instance, under certain cases, the quality of information does not influence cost. Our study has considered both risk averse and risk neutral approaches and provided insights. The insights obtained from our models are expected to help agencies reduce the cost of operations by choosing cost effective suppliers.

In this thesis we consider two research problems, namely, (i) language constructs for modelling stochastic programming (SP) problems and (ii) solution methods for processing instances of different classes of SP problems. We first describe a new design of an SP modelling system which provides greater extensibility and reuse. We implement this enhanced system and develop solver connections. We also investigate in detail the following important classes of SP problems: singlestage SP with risk constraints, two-stage linear and stochastic integer programming problems. We report improvements to solution methods for single-stage problems with second-order stochastic dominance constraints and two-stage SP problems. In both cases we use the level method as a regularisation mechanism. We also develop novel heuristic methods for stochastic integer programming based on variable neighbourhood search. We describe an algorithmic framework for implementing decomposition methods such as the L-shaped method within our SP solver system. Based on this framework we implement a number of established solution algorithms as well as a new regularisation method for stochastic linear programming. We compare the performance of these methods and their scale-up properties on an extensive set of benchmark problems. We also implement several solution methods for stochastic integer programming and report a computational study comparing their performance. The three solution methods, (a) processing of a single-stage problem with second-order stochastic dominance constraints, (b) regularisation by the level method for two-stage SP and (c) method for solving integer SP problems, are novel approaches and each of these makes a contribution to knowledge.

We study a natural generalization of the Λ-coalescent to a spatial continuum. We introduce the process, which is known as the Segregated Λ-coalescent, via its connections to the (non-spatial) Λ-coalescent and the Spatial Λ-Fleming-Viot process. The main new results contained in this thesis are as follows. The Segregated Λ-coalescent has a non-trivial construction which we present here in terms of stochastic flows. We describe the qualitative behaviour of the Segregated Λ-coalescent and compare it to the behaviour of the Λ-coalescent, showing in particular that the Segregated Λ-coalescent has an extra phase transition which is directly related to the introduction of space. We finish with some results concerning the rate at which the Segregated Λ-coalescent comes down from infinity.

This thesis deals with multi-stage stochastic linear programming and its ap- plictions in the portfolio selection problem. It presents several models of invest- ment planning, the emphasis is on the basic model with transaction costs and risk adjusted model for every investment level. Random returns entering the above models are modelled by the scenario trees which are generated using the moment- matching method. The thesis presents the optimal investment strategy for each model. It then examines distance of optimal values of objective functions in de- pendence on the nested distance of these generated trees. All calculations were performed using Mathematica software version 9. 1

Many biological processes can be described in terms of chemical reactions and diffusion. In this thesis, reaction-diusion mechanisms related to the formation of Turing patterns are studied. Necessary and sufficient conditions under which Turing instability occur is presented. Behaviour of Turing patterns is investigated with a use of deterministic approach, compartment-based stochastic simulation algorithm and molecular-based stochastic simulation algorithm.

This thesis deals with an issue of claims reserving for non-life insurance. The issue is approached in a sense of analytical calculation and stochastic modelling. First, Chain-ladder, Bornhuetter-Ferguson, Benktander-Hovinen and Cape-Cod method are introduced. In following chapters, we try to find related stochastic underlying models including Generalized linear models and Mack's distribution-free approaches, we analyze second moments of claims estimates for each of the methods and examine alternative Merz-Wüthrich approach to reserve risk measurement. At the end, bootstrap algorithm and estimates are suggested and simulation results are compared with analytic ones.

In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (SPDEs) by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.

This thesis describes the work of three research projects, the background research that motivated the work, and the resultant project findings. The three projects concerned: (i) Geometric stochastic resonance in a double cavity, (ii) Synchronisation of geometric stochastic resonance by a bi-harmonic drive, and (iii) Rectification of Brownian particles with oscillating radii in asymmetric corrugated channels. In the project 'Geometric stochastic resonance in a double cavity', we investigated synchronisation processes for the geometric stochastic resonance of particles diffusing across a porous membrane and subject to a periodic driving force. Non-interacting particle currents were driven through a symmetric membrane pore either parallel or perpendicular to the membrane. Then, harmonic mixing spectral current components were generated by the combined action of parallel and perpendicular drives. The role of the repulsive interaction of particles as a controlling factor with potential applications to the transport of colloids and biological molecules through narrow pores was also investigated. In 'Synchronisation of geometric stochastic resonance by a bi-harmonic drive', we simulated the stochastic dynamics of an elliptical particle using the Langevin equation. The particle was simultaneously driven by low and high frequency harmonic drives across a porous inter-cavity membrane. It was observed that the particle oscillated out of phase with the low frequency drive. This effect was due to the absolute negative mobility the particle would have exhibited if the low frequency drive had been replaced by a dc static force. It was also observed that the magnitude of this out-of-phase stochastic resonance depends on how the combined action of the driving forces and noise fluctuations affect the particle orientation, and as such was shown to be sensitive to the particle shape. This emphasises the importance of particle geometry, in addition to chamber geometry, to the realisation and optimisation of geometric stochastic resonance. In the project 'Rectification of Brownian particles with oscillating radii in asymmetric corrugated channels', we simulated the transport of a Brownian particle with an oscillating radius freely diffusing in an asymmetric corrugated channel over a range of driving forces for a series of temperatures and angular frequencies of radial oscillation. It was observed that there was a strong influence of self-oscillation frequency upon the average particle velocity. This effect can be used to control rectification of biologically active particles as well as for their separation according to their activity, for instance in the separation of living and dead cells. The background research is described in Chapter One and the research findings are described along with their related projects in Chapters Two and Three.