This thesis documents research undertaken on state and parameter estimation techniques for stochastic systems in a maintenance context. Two individual problem scenarios are considered. For the first scenario, we are concerned with complex systems and the research involves an investigation into the ability to identify and quantify the occurrence of fault injection during routine preventive maintenance procedures. This is achieved using an appropriate delay time modelling specification and maximum-likelihood parameter estimation techniques. The delay time model of the failure process is parameterised using objective information on the failure times and the number of faults removed from the system during preventive maintenance. We apply the proposed modelling and estimation process to simulated data sets in an attempt to recapture specified parameters and the benefits of improving maintenance processes are demonstrated for the particular example. We then extend the modelling of the system in a predictive manner and combine it with a stochastic filtering approach to establish an adaptive decision model. The decision model can be used to schedule the subsequent maintenance intervention during the course of an operational cycle and can potentially provide an improvement on fixed interval maintenance policies. The second problem scenario considered is that of an individual component subject to condition monitoring such as, vibration analysis or oil-based contamination. The research involves an investigation into techniques that utilise condition information that we assume is related stochastically to the underlying state of the component, taken here to be the residual life. The techniques that we consider are the proportional hazards model and a probabilistic stochastic filtering approach. We investigate the residual life prediction capabilities of the two techniques and construct relevant replacement decision models. The research is then extended to consider multiple indicators of condition obtained simultaneously at monitoring points. We conclude with a brief investigation into the use of stochastic filtering techniques in specific scenarios involving limited computational power and variable underlying relationships between the monitored information and the residual life.

This thesis comprises of five chapters. The first chapter gives a brief introduction on the Black-Scholes model and the Heston model that are used to develop a valuation framework for the European-style options. The numerical techniques that are applied to price the options-the fast Fourier transform (FFT) and the Monte Carlo simulation, are also introduced. The second chapter extends the existent literature on the pricing of power options. Under the Black-Scholes model and the Heston model, the equations for the characteristic functions for power options are solved, which are essential to effectively implement the FFT technique. In addition, under the Black-Scholes model, the Greeks for the power options, a power put-power call parity relationship, and a transformation relationship between vanilla options and power options are derived. Numerical experiments are run to evaluate the performance of the FFT approach. The results are compared to the approximations obtained from the Monte Carlo simulation. The third chapter extends the existent literature on the valuation of extendible options. The characteristic functions for extendible options are derived, under the Black-Scholes model and the Heston model, by separating the expectations into different segments. This chapter shows that the characteristic function for a one-time extendible option consists of a one-variate and a two-variate characteristic functions; hence this leads to a one-dimensional and a two-dimensional FFT application. Numerical experiments are run for an evaluation of the performance of the FFI. and the results are compared with the valuations obtained from the Monte Carlo simulation. The fourth chapter extends the second chapter by introducing a barrier feature to the power option. This is a combination of a power option and a barrier option which is named the power barrier opt ion. The closed-form solution is derived under the Black-Scholes model using the risk-neutral valuation approach. The final chapter gives the summaries of chapter two, three and four with suggestions for further development on the work in this thesis.

This thesis considers the application of locally stationary wavelet-based stochastic models to the analysis of image texture. In the first part we propose a test of stationarity for spatial data on a regular grid. This test is then incorporated into a segmentation framework in order to determine the number of textures contained within an image, a key feature to many texture segmentation approaches. These novel methods are subsequently applied to various texture analysis problems arising from work with an industrial collaborator. The second part of this thesis considers the modelling of the spectral structure of a non-stationary multivariate image, i.e. an image containing different colour channels. We propose a multivariate locally stationary wavelet-based modelling framework which permits a measure of dependence between pairs of channels. The performance of this modelling approach is then assessed using various colour texture examples encountered by an industrial collaborator.

Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult. One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization. In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis. We propose a powerful unifying paradigm that combines analytical and formal methods. Its applications vary from air traffic control to communication networks and healthcare systems. The stochastic hybrid system paradigm has an explosive development. This is because of its very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems. Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability analysis, model checking). These constitute the major motivations of our research. We investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural way. The concrete contribution of this thesis has four major milestones: 1. A new and a very general model for stochastic hybrid systems; 2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities; 3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated. 4. Considering the communication issue, we extend the modelling paradigm.

Dans cette thèse, nous développons des méthodes de contrôle pour stabiliser des systèmes quantiques en temps continu sous mesures quantiques non-destructives. En boucle ouverte, ces systèmes convergent vers un état propre de l'opérateur de mesure, mais l'état résultant est aléatoire. Le rôle du contrôle est de préparer un état prescrit avec une probabilité de un. Le nouvel élément pour atteindre cet objectif est l'utilisation d'un mouvement Brownien pour piloter les actions de contrôle. En utilisant la théorie stochastique de Lyapunov, nous montrons stabilité exponentielle globale du système en boucle fermés. Nous explorons aussi la syntèse du contrôle pour stabiliser un code correcteur d'erreurs quantiques en temps continu. Un autre sujet d'intérêt est l'implementation de contrôles efficacement calculables dans un contexte expérimental. Dans cette direction, nous proposons l'utilisation de contrôles et filtres qui calculent seulement les characteristiques classiques du système, correspondant a la base propre de l'opérateur de mesure. La formulation de dites filtres est importante pour adresser les problèmes de scalabilité du filtre posées par l'avancement des technologies quantiques. / In this thesis, we develop control methods to stabilize quantum systems in continuous-time subject to quantum nondemolition measurements. In open-loop such quantum systems converge towards a random eigenstate of the measurement operator. The role of feedback is to prepare a prescribed eigenstate with unit probability. The novel element to achieve this is the introduction of an exogenous Brownian motion to drive the control actions. By using standard stochastic Lyapunov techniques, we show global exponential stability of the closed-loop dynamics. We explore as well the design of the control layer for a quantum error correction scheme in continuous-time. Another theme of interest is towards the implementation of efficiently computable control laws in experimental settings. In this direction, we propose the use control laws and of reduced-order filters which only track classical characteristics of the system, corresponding to the populations on the measurement eigenbasis. The formulation of these reduced filters is important to address the scalability issues of the filter posed by the advancement of quantum technologies.