The main goal of this thesis is to develop Malliavin Calculus for Lévy processes. This will be achieved by using the representation property of square integrable functions; every Lévy process can be decomposed into a Wiener and Poisson random measure part. In the first part of the thesis we prove a chaos expansion for Lévy spaces. We can then define directional derivatives in the Wiener and Poisson Random measure directions, and reach an extended Clark-Ocone-Haussmann formula. Following that we define and study the properties of the adjoint operators of the directional derivatives - the Skorohod integrals in both directions. The theoretical part is concluded by studying under which conditions a solution of a stochastic differential equation belongs to the domain of the two directional derivatives. The last part of the thesis is devoted to the applications of the developed theory to finance. This will include the explicit calculation of minimal variance hedging strategies in incomplete markets and the computation of the Greeks.

In modern computer systems, workload varies at different times and locations. It is important to model the performance of such systems via workload models that are both representative and efficient. For example, model-generated workloads represent realistic system behaviour, especially during peak times, when it is crucial to predict and address performance bottlenecks. In this thesis, we model performance, namely throughput and delay, using adaptive models and discrete queues. Hidden Markov models (HMMs) parsimoniously capture the correlation and burstiness of workloads with spatiotemporal characteristics. By adapting the batch training of standard HMMs to incremental learning, online HMMs act as benchmarks on workloads obtained from live systems (i.e. storage systems and financial markets) and reduce time complexity of the Baum-Welch algorithm. Similarly, by extending HMM capabilities to train on multiple traces simultaneously it follows that workloads of different types are modelled in parallel by a multi-input HMM. Typically, the HMM-generated traces verify the throughput and burstiness of the real data. Applications of adaptive HMMs include predicting user behaviour in social networks and performance-energy measurements in smartphone applications. Equally important is measuring system delay through response times. For example, workloads such as Internet traffic arriving at routers are affected by queueing delays. To meet quality of service needs, queueing delays must be minimised and, hence, it is important to model and predict such queueing delays in an efficient and cost-effective manner. Therefore, we propose a class of discrete, processor-sharing queues for approximating queueing delay as response time distributions, which represent service level agreements at specific spatiotemporal levels. We adapt discrete queues to model job arrivals with distributions given by a Markov-modulated Poisson process (MMPP) and served under discriminatory processor-sharing scheduling. Further, we propose a dynamic strategy of service allocation to minimise delays in UDP traffic flows whilst maximising a utility function.

This thesis is concerned with rigorous computation of dynamical quantities. In particular, we provide rigorous computation of diffusion coefficients for uniformly expanding maps of the interval. Moreover, we provide a rigorous computational scheme for linear response and we apply it in the case of uniformly expanding circle maps. Our results have been implemented successfully on a computer. Examples are included to illustrate the computer implementation.

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This thesis addresses some aspects of the connection between convex analysis andoptimal stopping and control problems. The first chapter contains a summary of theoriginal contributions made in subsequent chapters. The second chapter uses elementary tools from convex analysis to establish anextension of the Legendre transformation. These results complement the results in[66] and are used to provide an alternative proof that Nash equilibria exist in optimalstopping games driven by diffusions. In the third chapter a ‘maximum principle’ for singular stochastic control is es-tablished using methods from convex analysis which is a generalisation of the firstorder conditions derived in [18]. This ‘maximum principle’ is used to show that thesolution to certain singular stochastic control problems can be expressed in termsof a family of associated optimal stopping problems. These results connect the firstorder conditions in [3] and the representation result originating in [5] to variationalanalysis. In particular, the Legendre transform is used to derive first order conditionsfor a class of constrained optimisation problems. Sections 2.1-2.4 and Example 30 have been accepted for publication to the ‘Journalof Convex Analysis’ as [75] subject to minor corrections. The suggested revision hasbeen implemented in this thesis.

Geotechnical analyses have traditionally followed a deterministic approach in which materials are modelled using representative property values. An alternative approach is to take into account the spatial variation, or heterogeneity, existing in all geomaterials. In this approach the material property is represented by a mean and standard deviation and by a definition of the spatial correlation. This leads to a stochastic-type analysis resulting in reliability assessments.Random finite element methods (RFEM) have been implemented incorporating spatial variability for a series of models. This variability is incorporated using random fields, which conform to the mean, standard deviation and spatial correlation of the modelled geomaterials. For each material the set of statistical parameters produces an infinite number of possible random fields; therefore a Monte Carlo approach is followed, by executing the FE analysis for hundreds of realizations of the random field. This stochastic approach is both time and memory exhaustive computationally, limiting domain sizes and significantly increasing run-times. With the advances in commercial computational resources, the demand for more accurate and 3D models has increased, further straining the computational resources required by the method. To reduce these effects the stages of the method have been parallelized; initially the FE analysis, then the Monte Carlo framework and finally the random field generation. This has led to increases in the executable domain sizes and reductions in the run-times of the method. The new parallel codes have been used to analyse large-scale 3D slope reliability problems, which previously could not be undertaken in a serial environment. These computations have demonstrated the effectiveness of the new implementation, as well as adding confidence in the conclusions of previous research carried out on smaller 3D domains.

Classical multivariate analysis techniques such as principal components analysis (PCA), canonical correlation analysis (CCA) and discriminant analysis (DA) can be badly affected when extreme outliers are present. The purpose of this thesis is to present new robust versions of these methods. Our approach is based on the following observation: the classical approaches to PCA, CCA and DA can all be interpreted as operations on a Gaussian likelihood function. Consequently, PCA, CCA and DA can be robustified by replacing the Gaussian likelihood with a Cauchy likelihood. The performance of the Cauchy version of each of these procedures is studied in detail both theoretically, through calculation of the relevant influence function, and numerically, through numerous examples involving real and simulated data. Our results demonstrate that the new procedures have good robustness properties which are certainly far superior to these of the classical versions.

The Brownian excursion is defined as a standard Brownian motion conditioned on starting and ending at zero and staying positive in between. The first part of the thesis deals with functionals of the Brownian excursion, including first hitting time, last passage time, maximum and the time it is achieved. Our original contribution to knowledge is the derivation of the joint probability of the maximum and the time it is achieved. We include a financial application of our probabilistic results on Parisian default risk of zero-coupon bonds. In the second part of the thesis the Parisian, occupation and local time of a drifted Brownian motion is considered, using a two-state semi-Markov process. New versions of Parisian options are introduced based on the probabilistic results and explicit formulae for their prices are presented in form of Laplace transforms. The main focus in the last part of the thesis is on the joint probability of Parisian and hitting time of Brownian motion. The difficulty here lies in distinguishing between different scenarios of the sample path. Results are achieved by the use of infinitesimal generators on perturbed Brownian motion and applied to innovative equity exotics as generalizations of the Barrier and Parisian option with the advantage of being highly adaptable to investors’ beliefs in the market.