The thesis focuses on the identification of linear systems and those non-linear systems which can be represented with purely linear dynamics and a zero memory nonlinearity in cascade. With the systems subjected to stationary Gaussian processes or periodic excitations and with or without external disturbances at the output, the linear dynamics are estimated as non-parametric or parametric models and any non-linearity is characterised through the form of sign~ls appearing at its input and output. The main research of the first part of the thesis is concerned with non-parametric identification of linear systems with finite time stationary white Gaussian data, or finite gain and phase response measurements. The sources of errors leading to the uncertainty of the frequency response function of a system are identified and it is shown that there is a limit to the reduction in the variance when wind~wing the measured data and block overlap is employ~d. The direct use of expressions relating phase and gain responses lead to inaccurate results, and modifications to the functions and extrapolation methods are developed to give significant improvement in accuracy. The second part involves non-linear system identification. By using a sinusoidal excitation, it is shown how the phase of a harmonic at the output relative to the input can be used to deduce the position of the non-linearity in relation to the linear dynamics. Two procedures are developed to identify the dynamics and form of non-linearity in the system. Further, it is shown how among a class of discrete time linear models, the auto-regressive moving average with exogenous input (ARMAX) best describes the linear dynamics of a Hammerstein system while a Box-Jenkins (BJ) best describes the linear dynamics of a Wiener system, in a mean square sense. The last part of the thesis gives a review for periodic perturbation signal design. Graphical user interfaces were developed to ease the generation of pseudo random and multilevel multiharmonic signals.

We introduce a new, fast and accurate method to calculate prices and sensitivities of European vanilla and digital options under the Variance Gamma model. For near at-the-money options of short maturity, our method is much faster than those based on discretization and truncation of the inverse Fourier transform integral (iFT method). We show that the results calculated with our method agree with those obtained with the iFT algorithm using very long and fine grids. Taking the results of our method as a benchmark, we show that the parabolic modification of the iFT method (Boyarchenko and Levendorskiĭ, 2012) is much more efficient than the standard (flat) version. Based on this conclusion, we consider an approach which uses a combination of backward induction and parabolic iFT to price discretely monitored barrier options, as well as credit default swaps, under wide classes of Lévy models. At each step of backward induction, we use piece-wise polynomial interpolation and parabolic iFT, which allows for efficient error control. We derive accurate recommendations for the choice of parameters of the numerical scheme, and produce numerical examples showing that oversimplified prescriptions in other methods can result in large errors.

The sequential Monte Carlo (SMC) methodology is a family of Monte Carlo methods that processes information sequentially. It has shown to be able to solve a large class of highly complex inference and optimization problems that can be formulated as stochastic dynamic systems. By recursively generating random samples of the state variables of the dynamic systems, SMC adapts flexibly to the dynamics of the underlying stochastic systems. It opens up new frontiers for cross-fertilization between statistical science and many application areas.

This thesis makes two extensions to the standard stochastic approximation framework in order to study learning algorithms in different environments. In particular, the aim of this has been to study fictitious play and stochastic fictitious play in more complex frameworks than the usual, normal form game environment. However, these stochastic approximation frameworks are also utilised in other applications in this thesis. A new two-timescale asynchronous stochastic approximation framework with set-valued updates is presented, which extends the previous work in this area by Konda and Borkar (2000). Using this approach a two-timescales learning algorithm is produced for discounted reward Markov decision processes and, similarly, fictitious play is studied in stochastic games. In the second half of this thesis an update to the existing abstract stochastic approximation framework based on the asymptotic pseudo-trajectory approach of Benaim (1999), is presented. Importantly, in this thesis criteria are given to control the noise term associated with this abstract stochastic approximation for certain useful Banach spaces. The logit best response dynamic has previously been studied in continuous action games by Lahkar and Riedel (2013). Their existence results are extended for the N-player case and a convergence result is proved for two-player zero-sum games with continuous actions sets. Stochastic fictitious play is then studied using abstract stochastic approximation and is shown to converge to a logit equilibrium strategy in two-player zero-sum games with continuous action sets . The final chapter of this thesis studies Newton's algorithm, which can be used as a computationally efficient method for estimating a mixing density in a mixture model. Tokdar et al. (2009) give certain conditions for this algorithm to converge to the true mixing density when the parameter space is an uncountable subset of R. One of their assumptions is removed and the convergence result strengthened to produce an alternative consistency result for Newton's Algorithm.

Three optimal dividend models are considered for which the underlying risk process is a spectrally negative Levy process. The first one concerns the classical dividends problem of de Finetti for which we give sufficient conditions under which the optimal strategy is of barrier type. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.

We consider several first passage problems for stable processes, giving explicit formulas for hitting distributions, hitting probabilities and potentials of stable processes killed at first passage. Our principal tools are the Lamperti representation of positive self-similar Markov processes and the Wiener-Hopf factorisation of Levy processes. As part of the proof apparatus, we introduce a new class of Levy processes with explicit Wiener- Hopf factorisation, which appear repeatedly in Lamperti representations derived from stable processes. We also apply the Lamperti-Kiu representation of real self-similar Markov processes and obtain results on the exponential functional of Markov additive processes, in order to find the law of the first time at which a stable process reaches the origin.

In this thesis we will investigate inference processes for predicate languages. The main question we are concerned with in this thesis is how to choose a probability function amongst those that satisfy a certain knowledge base. This question has been extensively studied for propositional logic and we shall investigate it for first order languages. We will first study the generalisation of Minimum Distance. MD. and Centre of Mass. CMco inference processes to unary predicate languages and then we will investigate the generalisations of itie Maximum Entropy inference process to general polyadic languages. For the case of the Maximum Entropy inference process we will study and compare two generalisations. the BP-method and the W-method. We will show that the two methods agree for the unary and :E, knowledge bases and we conjecture that the result holds for the II, knowledge bases too. We shall show that neither of these generalisations for the Maximum Entropy inference process is universally well defined for a first order language and we shall study some of the problems associated with generalising this inference process to polyadic languages.

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