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1
Rad, Soroush Rafiee. "Inference processes for probabilistic first order languages." Thesis, University of Manchester, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.506857.
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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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Loeffen, Ronnie Lambertus. "Stochastic control for spectrally negative Lévy processes." Thesis, University of Bath, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.505712.
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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.
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Watson, Alexander Rhys. "Stable process." Thesis, University of Bath, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.608335.
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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.
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Khalil, Hassan Kamel. "Particle Approximation in Stochastic Filtering." Thesis, University of Bristol, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.492654.
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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.
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Perkins, Steven. "Advanced stochastic approximation frameworks and their applications." Thesis, University of Bristol, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.601159.
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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.
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Srisuma, Sorawoot. "Essays on semiparametric estimation of Markov decision processes." Thesis, London School of Economics and Political Science (University of London), 2010. http://etheses.lse.ac.uk/2371/.
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Dynamic models of forward looking agents, whose goal is to maximize expected in-tertemporal payoffs, are useful modelling frameworks in economics. With an exception of a small class of dynamic decision processes, the estimation of the primitives in these models is computationally burdensome due to the presence of the value functions that has no closed form. We follow a popular two-step approach which estimates the functions of interest rather than use direct numerical approximation. The first chapter, joint with Oliver Linton, considers a class of dynamic discrete choice models that contain observable continuously distributed state variables. Most papers on the estimation of dynamic discrete choice models assume that the observable state variables can only take finitely many values. We show that the extension to the infinite dimensional case leads to a well-posed inverse problem. We derive the distribution theory for the finite and the infinite dimensional parameters. Dynamic models with continuous choice can sometimes avoid the numerical issues related to the value function through the use of Euler's equation. The second chapter considers models with continuous choice that do not necessarily belong to the Euler class but frequently arise in applied problems. In this chapter, a class of minimum distance estimators is proposed, their distribution theory along with the infinite dimensional parameters of the decision models are derived. The third chapter demonstrates how the methodology developed for the discrete and continuous choice problems can be adapted to estimate a variety of other dynamic models. The final chapter discusses an important problem, and provides an example, where some well-known estimation procedures in the literature may fail to consistently estimate an identified model. The estimation methodologies I propose in the preceding chapters may not suffer from the problems of this kind.
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Widanage, Widanalage Dhammika. "The Effects of Distortions on LinearSystem Identification and Non-linearCharacterisation." Thesis, University of Warwick, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.502124.
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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.
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de, Innocentis Marco. "Pricing discretely monitored barrier options and credit default swaps under Lévy processes." Thesis, University of Leicester, 2013. http://hdl.handle.net/2381/27912.
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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.
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Hamilton, Emily. "Use of extreme value theory for making statistical inference about endpoints of distributions, with applications in global optimization and meteorology." Thesis, Cardiff University, 2008. http://orca.cf.ac.uk/54789/.
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We use extreme value theory to make statistical inference about the endpoint of distributions. First we compare estimators of the endpoint of several distributions, including a distribution that appears in problems of global optimization. These estimators use a fixed number of order statistics (k) from a sample of fixed size (n). Two of the estimators investigated are the optimal linear estimator and the maximum likelihood estimator. We find that the optimal linear estimator often outperforms the maximum likelihood estimator. We next investigate how the order statistics change as sample size increases. In order to do this, we define record times: the sample size at which the set of k smallest order statistics changes. We give the distributions of several statistics related to order statistics and record times, in particular we show that records occur according to a nonhomogeneous Poisson process. We show that order statistics can be modeled using a Markov chain, and use this Markov chain to investigate estimators of the endpoint of a distribution. Two additional estimators are derived and investigated using the Markov chain model. Finally, we consider a meteorological application of extreme value theory. In particular, we estimate the maximum and minimum sea level at several ports in the Netherlands. This is done using a combination of record theory, singular spectrum decomposition and known estimators of the endpoint of a distribution.
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Savani, Vippal. "Statistical inference for negative binomial processes with applications to market research." Thesis, Cardiff University, 2006. http://orca.cf.ac.uk/56126/.
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The negative binomial distribution (NBD) and negative binomial processes have been used as natural models for events occurring in fields such as accident proneness accidents and sickness market research insurance and risk theory. The fitting of negative binomial processes in practice has mainly focussed on fitting the one-dimensional distribution, namely the NBD, to data. In practice, the parameters of the NBD are usually estimated by using inefficient moment based estimation methods due to the ease in estimating moment based estimators in comparison to maximum likelihood estimators. This thesis develops efficient moment based estimation methods for estimating parameters of the NBD that can be easily implemented in practice. These estimators, called power method estimators, are almost as efficient as maximum likelihood estimators when the sample is independent and identically distributed. For dependent NBD samples, the power method estimators are more efficient than the commonly used method of moments and zero term method estimators. Fitting the one-dimensional marginal distribution of negative binomial processes to data gives partial information as to the adequacy of the process being fitted. This thesis further develops methods of statistical inference for data generated by negative binomial processes by comparing the dynamical properties of the process to the dynamical properties of data. For negative binomial autoregressive processes, the dynamical properties may be checked by using the autocorrelation function. The dynamical properties of the gamma Poisson process are considered by deriving the asymptotic covariance and correlation structures of estimators and functionals of the gamma Poisson process and verifying these structures against data. The adequacy of two negative binomial processes, namely the gamma Poisson process and the negative binomial first-order autoregressive process, as models for consumer buying behavior are considered. The models are fitted to market research data kindly provided by ACNielsen BASES.
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Al, Azemi Fares M. M. S. "A pair of explicitly solvable impulse control problems." Thesis, London School of Economics and Political Science (University of London), 2010. http://etheses.lse.ac.uk/2773/.
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This thesis is concerned with the formulation and the explicit solution of two impulse stochastic control problems that are motivated by applications in the area of sequential investment decisions. Each of the two problems considers a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Ito diffusion. In the first of the two problems, the control that can be applied to the system takes the form of one-sided impulsive action, and the associated objective is to maximise a performance criterion that rewards high values of the utility derived from the system's controlled state and penalises the expenditure of any control effort. Potential applications of this model arise in the area of real options where one has to balance the sunk costs incurred by investment against their resulting uncertain cashflows. The second model is concerned with the so-called buy-low and sell-high investment strategies. In this context, an investor aims at maximising the expected discounted cash-flow that can be generated by sequentially buying and selling one share of a given asset at fixed transaction costs. Both of the control problems are solved in a closed analytic form and the associated optimal control strategies are completely characterised. The main results are illustrated by means of special cases that arise when the uncontrolled system dynamics are a geometric Brownian motion or a mean-reverting square-root process such as the one in the Cox-Ingersoll-Ross interest rate model.
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Cardoso, Rui Manuel Rodrigues. "Numerical algorithms for the calculation of finite time ruin probabilities in generalisations of the classical risk model." Thesis, Heriot-Watt University, 2004. http://hdl.handle.net/10399/343.
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Anzures-Cabrera, Judith. "Survival analysis : competing risks, truncation and immunes." Thesis, University of Warwick, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.413431.
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Sheehan, Marcus. "Some aspects of tree-indexed processes." Thesis, University of Warwick, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.439542.
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Myatt, Darren Robert. "Analysis of stochastic diffusion search and its application to robust estimation." Thesis, University of Reading, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.440105.
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Matsikis, Iakovos. "High gain control of stochastic differential equations." Thesis, University of Exeter, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403248.
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Hunter, Kieran. "Optimal generalised measurement strategies." Thesis, University of Strathclyde, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.401315.
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O'Donnell, David. "Bayesian inference for graphical Gaussian and conditional Gaussian models." Thesis, University of Southampton, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433936.
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Owen, Nathan Edward. "A comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments." Thesis, University of Exeter, 2017. http://hdl.handle.net/10871/29296.
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Computer simulation of real world phenomena is now ubiquitous in science, because experimentation in the field can be expensive, time-consuming, or impossible in practice. Examples include climate science, where future climate is examined under global warming scenarios, and cosmology, where the evolution of galaxies is studied from the beginning of the universe to present day. Combining complex mathematical models and numerical procedures to solve them in a computer program, these simulators are computationally expensive, in that they can take months to complete a single run. The practice of using a simulator to understand reality raises some interesting scientific questions, and there are many sources of uncertainty to consider. For example, the discrepancy between the simulator and the real world process. The field of uncertainty quantification is concerned with the characterisation and reduction of all uncertainties present in computational and real world problems. A key bottleneck in any uncertainty quantification analysis is the cost of evaluating the simulator. The solution is to replace the expensive simulator with a surrogate model, which is computationally faster to run, and can be used in subsequent analyses. Polynomial chaos and Gaussian process emulation are surrogate models developed independently in the engineering and statistics communities respectively over the last 25 years. Despite tackling similar problems in the field, there has been little interaction and collaboration between the two communities. This thesis provides a critical comparison of the two methods for a range of criteria and examples, from simple test functions to simulators used in industry. Particular focus is on the approximation accuracy of the surrogates under changes in the size and type of the experimental design. It is concluded that one method does not unanimously outperform the other, but advantages can be gained in some cases, such that the preferred method depends on the modelling goals of the practitioner. This is the first direct comparison of polynomial chaos and Gaussian process emulation in the literature. This thesis also proposes a novel methodology called probabilistic polynomial chaos, which is a hybrid of polynomial chaos and Gaussian process emulation. The approach draws inspiration from an emerging field in scientific computation known as probabilistic numerics, which treats classical numerical methods as statistical inference problems. In particular, a probabilistic integration technique called Bayesian quadrature, which employs Gaussian process emulators, is applied to a traditional form of polynomial chaos. The result is a probabilistic version of polynomial chaos, providing uncertainty information where the simulator has not yet been run.
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Turner, Lisa. "Inference and decision making in large weakly dependent graphical models." Thesis, Lancaster University, 2017. http://eprints.lancs.ac.uk/88194/.
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This thesis considers the problem of searching a large set of items, such as emails, for a small subset which are relevant to a given query. This can be implemented in a sequential manner – whereby knowledge from items that have already been screened is used to assist in the selection of subsequent items to screen. Often the items being searched have an underlying network structure. Using the network structure and a modelling assumption that relevant items and participants are likely to cluster together can greatly increase the rate of screening relevant items. However, inference in this type of model is computationally expensive. In the first part of this thesis, we show that Bayes linear methods provide a natural approach to modelling this data. We develop a new optimisation problem for Bernoulli random variables, called constrained Bayes linear, which has additional constraints incorporated into the Bayes linear optimisation problem. For non-linear relationships between the latent variable and observations, Bayes linear will give a poor approximation. We propose a novel sequential Monte Carlo method for sequential inference on the network, which better copes with non-linear relationships. We give a method for simulating the random variables based upon the Bayes linear methodology. Finally, we look at the effect the ordering of the random variables has on the joint probability distribution of binary random variables, when they are simulated using this proposed Bayes linear method.
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Ogrodnik, Marcel Bogdan. "Markovian rough paths." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/51500.
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The accumulated local p-variation functional, originally presented by Cass et al. (2013), arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local p-variation functional typically decay much faster than the tails of classical p-variation. This observation has been decisive, e.g. for problems involving Malliavin calculus for Gaussian rough paths as illustrated in the work by Cass et al. (2015). All of the examples treated so far have been in this Gaussian setting, that contains a great deal of additional structure. In this paper we work in the context of Markov processes on a locally compact Polish space E, which are associated to a class of Dirichlet forms. In this general framework, we first prove a better-than-exponential tail estimate for the accumulated local p-variation functional derived from the intrinsic metric of this Dirichlet form. By then specialising to a class of Dirichlet forms on the step-⌊p⌋ free nilpotent group, which are subelliptic in the sense of Fefferman-Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes, but also goes beyond, the examples studied by Friz and Victoir (2008). We comment on the significance of these estimates to recent results, including the results of Hao (2014) and Chevyrev and Lyons (2015).
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Patacchini, Francesco Saverio. "A variational and numerical study of aggregation-diffusion gradient flows." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/50180.
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This thesis is dedicated to the variational and numerical study of a particular class of continuity equations called aggregation-diffusion equations. They model the evolution of a continuum body whose total mass is conserved in time, undergoing up to three distinct phenomena: diffusion, confinement and aggregation. Diffusion describes the motion of the body’s particles from crowded regions of space to sparser ones; confinement results from an external potential field independent of the mass distribution of the body; and aggregation describes the nonlocal particle interaction within the body. Due to this wide range of effects, aggregation-diffusion equations are encountered in a large variety of applications coming from, among many others, porous medium flows, granular flows, crystallisation, biological swarming, bacterial chemotaxis, stellar collapse, and economics. An aggregation-diffusion equation has the very interesting and rich mathematical property of being the gradient flow for some energy functional on the space of probability measures, which formally means that any solution evolves so as to decrease this energy every time as much as possible. In this thesis we exploit this gradient-flow structure of aggregation-diffusion equations in order to derive properties of solutions and approximate them by discrete particles. We focus on two main aspects of aggregation-diffusion gradient flows: the variational analysis of the pure aggregation equation, i.e., the study of minimisers of the energy when only nonlocal aggregation effects are present; and the particle approximation of solutions, especially when only diffusive effects are taken into account. Regarding the former aspect, we prove that minimisers exist, enjoy some regularity, are supported on sets of specific dimensionality, and can be approximated by finitely supported discrete minimisers. Regarding the latter aspect, we illustrate theoretically and numerically that diffusion can be interpreted at the discrete level by a deterministic motion of particles preserving a gradient-flow structure.
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Wang, C. "On asymptotic stability of stochastic differential equations with delay in infinite dimensional spaces." Thesis, University of Liverpool, 2017. http://livrepository.liverpool.ac.uk/3007651/.
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In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion.
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Frot, Benjamin. "Graphical model selection for Gaussian conditional random fields in the presence of latent variables : theory and application to genetics." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:0a6799ed-fca1-48b2-89cd-ad6f2c0439af.
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The task of performing graphical model selection arises in many applications in science and engineering. The field of application of interest in this thesis relates to the needs of datasets that include genetic and multivariate phenotypic data. There are several factors that make this problem particularly challenging: some of the relevant variables might not be observed, high-dimensionality might cause identifiability issues and, finally, it might be preferable to learn the model over a subset of the collection while conditioning on the rest of the variables, e.g. genetic variants. We suggest addressing these problems by learning a conditional Gaussian graphical model, while accounting for latent variables. Building on recent advances in this field, we decompose the parameters of a conditional Markov random field into the sum of a sparse and a low-rank matrix. We derive convergence bounds for this novel estimator, show that it is well-behaved in the high-dimensional regime and describe algorithms that can be used when the number of variables is in the thousands. Through simulations, we illustrate the conditions required for identifiability and show that this approach is consistent in a wider range of settings. In order to show the practical implications of our work, we apply our method to two real datasets and devise a metric that makes use of an independent source of information to assess the biological relevance of the estimates. In our first application, we use the proposed approach to model the levels of 39 metabolic traits conditional on hundreds of genetic variants, in two independent cohorts. We find our results to be better replicated across cohorts than the ones obtained with other methods. In our second application, we look at a high-dimensional gene expression dataset. We find that our method is capable of retrieving as many biologically relevant gene-gene interactions as other methods while retrieving fewer irrelevant interaction.
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Eckhoff, Maren. "Superprocesses and Large-Scale Networks." Thesis, University of Bath, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.675692.
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The main theme of this thesis is the use of the branching property in the analysis of random structures. The thesis consists of two self-contained parts. In the first part, we study the long-term behaviour of supercritical superdiffusions and prove the strong law of large numbers. The key tools are spine and skeleton decompositions, and the analysis of the corresponding diffusions and branching particle diffusions. In the second part, we consider preferential attachment networks and quantify their vulnerability to targeted attacks. Despite the very involved global topology, locally the network can be approximated by a multitype branching random walk with two killing boundaries. Our arguments exploit this connection.
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Pagett, Steven. "Fragmentation-coalescence processes : theory and applications." Thesis, University of Bath, 2017. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.715297.
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The main objects of study in this thesis are fragmentation-coalescence processes, where particles are grouped into clusters and evolve by either joining together, to form larger clusters, or splitting apart, to form smaller clusters. The focus is on the number of these clusters and the distribution of their sizes. In particular, we show for a certain class of processes defined on a finite system that there is convergence in the thermodynamic limit to an infinite system. For a second class of processes we show there is a phase transition between regimes where the number of clusters has an entrance law from ∞ or not.
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Kirsebom, Maxim Solund. "Extreme value theory for group actions on homogeneous spaces." Thesis, University of Bristol, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.664970.
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In this thesis we study extreme value theory for random walks as well as one-parameter actions on homogeneous spaces. In both cases we investigate the limiting distributions for the maximum of an observable evaluated along a trajectory of the system. In particular we are going to consider asymptotic distributions for closest distance returns to a given point· and tor maximal excursions to the cusp. For closest returns on the torus we establish an exact extreme value distribution while for other cases we obtain estimates on the extreme value distributions for sparse sequences. For random walks we also obtain logarithm laws for the maximum. Finally we look into the extreme value statistics of exceedances of high levels in these settings. For the closest returns we establish convergence to a Poisson process for the point process of exceedances. In other cases we obtain estimates on the limiting distribution of the k'th largest maximum for sparse sequences.
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Bryden, Alan. "Stability issues in the numerical solution of stochastic differential equations." Thesis, University of Strathclyde, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.401344.
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Masuadi, E. "Non-parametric competing risks with multivariate frailty models." Thesis, Oxford Brookes University, 2013. http://radar.brookes.ac.uk/radar/items/e828e4da-de08-2f34-37b0-8cc3bbaf7150/1.
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This research focuses on two theories: (i) competing risks and (ii) random eect (frailty) models. The theory of competing risks provides a structure for inference in problems where cases are subject to several types of failure. Random eects in competing risk models consist of two underlying distributions: the conditional distribution of the response variables, given the random eect, depending on the explanatory variables each with a failure type specic random eect; and the distribution of the random eect. In this situation, the distribution of interest is the unconditional distribution of the response variable, which may or may not have a tractable form. The parametric competing risk model, in which it is assumed that the failure times are coming from a known distribution, is widely used such as Weibull, Gamma and other distributions. The Gamma distribution has been widely used as a frailty distribution, perhaps due to its simplicity since it has a closed form expression of the unconditional hazard function. However, it is unrealistic to believe that a few parametric models are suitable for all types of failure time. This research focuses on a distribution free of the multivariate frailty models. Another approach used to overcome this problem is using nite mixture of parametric frailty especially those who have a closed form of unconditional survival function. In addition, the advantages and disadvantages of a parametric competing risk models with multivariate parametric and/or non-parametric frailty (correlated random eects) are investigated. In this research, four main models are proposed: rst, an application of a new computation and analysis of a multivariate frailty with competing risk model using Cholesky decomposition of the Lognormal frailty. Second, a correlated Inverse Gaussian frailty in the presence of competing risks model. Third, a non-parametric multivariate frailty with parametric competing risk model is proposed. Finally, a simulation study of nite mixture of Inverse Gaussian frailty showed the ability of this model to t dierent frailty distribution. One main issue in multivariate analysis is the time it needs to t the model. The proposed non-parametric model showed a signicant time decrease in estimating the model parameters (about 80% less time compared the Log-Normal frailty with nested loops). A real data of recurrence of breast cancer is used as the applications of these models.
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Riedler, Martin Georg. "Spatio-temporal stochastic hybrid models of biological excitable membranes." Thesis, Heriot-Watt University, 2011. http://hdl.handle.net/10399/2484.
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A large number of biological systems are intrinsically random, in particular, biological excitable membranes, such as neuronal membranes, cardiac tissue or models for calcium dynamics. The present thesis is concerned with hybrid stochastic models of spatio-temporal dynamics of biological excitable membranes using Piecewise Deterministic Markov Processes (PDMPs). This class of processes allows a precise mathematical description of the internal noise structure of excitable membranes. Overall the aim of the thesis is two-fold: On the one hand, we establish a general hybrid modelling framework for biological excitable membranes and, on the other hand, we are interested in a general advance of PDMP theory which the former necessitates. Regarding the first aim we exemplify the modelling framework on the classical Hodgkin-Huxley model of a squid giant axon. Regarding the latter we present a general PDMP theory incorporating spatial dynamics and present tools for their analysis. Here we focus on two aspects. Firstly, we consider the approximation of PDMPs by deterministic models or continuous stochastic processes. To this end we derive as general theoretical tools a law of large numbers for PDMPs and martingale central limit theorems. The former establishes a connection of stochastic hybrid models to deterministic models given, e.g., by systems of partial differential equations. Whereas the latter connects the stochastic fluctuations in the hybrid models to diffusion processes. Furthermore, these limit theorems provide the basis for a general Langevin approximation to PDMPs, i.e., certain stochastic partial differential equations that are expected to be similar in their dynamics to PDMPs. Secondly, we also address the question of numerical simulation of PDMPs. We present and analyse the convergence in the pathwise sense of a class of simulation methods for PDMPs in Euclidean space.
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Janiszewski, Szymon Pawel. "Optimization problems in discrete and continuous time." Thesis, University of Hull, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396087.
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Rao, V. A. P. "Markov chain Monte Carlo for continuous-time discrete-state systems." Thesis, University College London (University of London), 2012. http://discovery.ucl.ac.uk/1349490/.
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A variety of phenomena are best described using dynamical models which operate on a discrete state space and in continuous time. Examples include Markov (and semi-Markov) jump processes, continuous-time Bayesian networks, renewal processes and other point processes. These continuous-time, discrete-state models are ideal building blocks for Bayesian models in fields such as systems biology, genetics, chemistry, computing networks, human-computer interactions etc. However, a challenge towards their more widespread use is the computational burden of posterior inference; this typically involves approximations like time discretization and can be computationally intensive. In this thesis, we describe a new class of Markov chain Monte Carlo methods that allow efficient computation while still being exact. The core idea is an auxiliary variable Gibbs sampler that alternately resamples a random discretization of time given the state-trajectory of the system, and then samples a new trajectory given this discretization. We introduce this idea by relating it to a classical idea called uniformization, and use it to develop algorithms that outperform the state-of-the-art for models based on the Markov jump process. We then extend the scope of these samplers to a wider class of models such as nonstationary renewal processes, and semi-Markov jump processes. By developing a more general framework beyond uniformization, we remedy various limitations of the original algorithms, allowing us to develop MCMC samplers for systems with infinite state spaces, unbounded rates, as well as systems indexed by more general continuous spaces than time.
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Leuwattanachotinan, Charnchai. "Model fitting of a two-factor arbitrage-free model for the term structure of interest rates using Markov chain Monte Carlo." Thesis, Heriot-Watt University, 2011. http://hdl.handle.net/10399/2425.
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In this thesis we use Markov chain Monte Carlo (MCMC) simulation to calibrate a two-factor arbitrage-free model for the term structure of interest rates which is proposed by Cairns (2004a) based on the positive-interest framework (Flesaker and Hughston, 1996). The model is a time-homogeneous model driven by latent state variables which follow a two-dimensional Ornstein-Uhlenbeck process. A number of MCMC algorithms are developed and employed for estimating both model parameters and latent variables where simulated data are used in the first place in order to validate the algorithms and ensure that they can result in reasonable and reliable estimates before using UK market data. Once the posterior estimates are obtained, we next investigate goodness of fit of the model and eventually assess the impact of parameter uncertainty on the forecasting of yield curves in which the achieved MCMC output can be used directly. Additionally, the developed algorithm is also applied for estimating the two-factor Vasicek term structure model for comparison. We conclude that our algorithms work reasonably well for estimating the Cairns term structure model. The model is then fitted to UK Strips data, and it found to produce reasonable fits for medium- and long-term yields, but we also conclude that some improvement may be required for the short-end of the yield curves.
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Adamu, Iyabo Ann. "Numerical approximation of SDEs and stochastic Swift-Hohenberg equation." Thesis, Heriot-Watt University, 2011. http://hdl.handle.net/10399/2460.
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We consider the numerical approximation of stochastic differential equations interpreted both in the It^o and Stratonovich sense and develop three stochastic time-integration techniques based on the deterministic exponential time differencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary differential equations (SODEs) and they are referred to as the stochastic exponential time differencing schemes, SETD0 and SETD1. The third numerical scheme is a new numerical method we propose for the simulations of Stratonovich SODEs. We call this scheme, the Exponential Stratonovich Integrator (ESI). We investigate numerically the convergence of these three numerical methods, in addition to three standard approximation schemes and also compare the accuracy and efficiency of these schemes. The effect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time differencing scheme (SETD0) for parabolic stochastic partial differential equations (SPDEs) with infinite-dimensional additive noise and one-dimensional multiplicative noise. We obtain a strong error temporal estimate of O(¢tµ + ²¢tµ + ²2¢t1=2) for SPDEs forced with a one-dimensional multiplicative noise and also obtain a strong error temporal estimate of O(¢tµ + ²2¢t) for SPDEs forced with an infinite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the effects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equation, and allow the control parameter to fluctuate. Numerical simulations show a shift in the pinning region with multiplicative noise.
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Kollar, Jozef. "Optimal Martingale measures and hedging in models driven by Levy processes." Thesis, Heriot-Watt University, 2011. http://hdl.handle.net/10399/2508.
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Our research falls into a broad area of pricing and hedging of contingent claims in incomplete markets. In the rst part we introduce the L evy processes as a suitable class of processes for nancial modelling purposes. This in turn causes the market to become incomplete in general and therefore the martingale measure for the pricing/hedging purposes has to be chosen by introducing some subjective criteria. We study several such criteria in the second section for a general stochastic volatility model driven by L evy process, leading to minimal martingale measure, variance-optimal, or the more general q-optimal martingale measure, for which we show the convergence to the minimal entropy martingale measure for q # 1. The martingale measures studied in the second section are put to use in the third section, where we consider various hedging problems in both martingale and semimartingale setting. We study locally risk-minimization hedging problem, meanvariance hedging and the more general p-optimal hedging, of which the meanvariance hedging is a special case for p = 2. Our model allows us to explicitly determine the variance-optimal martingale measure and the mean-variance hedging strategy using the structural results of Gourieroux, Laurent and Pham (1998) extended to discontinuous case by Arai (2005a). Assuming a Markovian framework and appealing to the Feynman-Kac theorem, the optimal hedge can be found by solving a three-dimensional partial integrodi erential equation. We illustrate this in the last section by considering the variance-optimal hedge of the European put option, and nd the solution numerically by applying nite di erence method.
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Livingstone, S. J. "Some contributions to the theory and methodology of Markov chain Monte Carlo." Thesis, University College London (University of London), 2016. http://discovery.ucl.ac.uk/1473910/.
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The general theme of this thesis is developing a better understanding of some Markov chain Monte Carlo methods. We review the literature in Chapters 1-4, including a short discussion of geometry in Markov chain Monte Carlo. In Chapter 5 we consider Langevin diffusions. First, a new class of these are derived in which the volatility is made position-dependent, using tools from stochastic analysis. Second, a complementary derivation is given, here using tools from Riemannian geometry. We hope that this work will help develop understanding of the geometric perspective among statisticians. Such derivations have been attempted previously, but solutions were not correct in general. We highlight these issues in detail. In the final part discussion is given on the use of these objects in Markov chain Monte Carlo. In Chapter 6 we consider a Metropolis-Hastings method with proposal kernel N(x,hV(x)), where x is the current state. After reviewing instances in the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of V(x) can change these compared to the Random Walk Metropolis case N(x,hS), for better or worse. In higher dimensions we show that judicious choice of V(x) can produce a geometrically converging chain when probability concentrates on an ever narrower ridge as |x| grows, something which is not true for the Random Walk Metropolis. In Chapter 7 we discuss stability of Hamiltonian Monte Carlo. For a fixed integration time we establish conditions for irreducibility and geometric ergodicity. Some results are confined to one dimension, and some further to a reference class of distributions. We find that target distributions with tails that are in between Exponential and Gaussian are needed for geometric ergodicity. Next we consider changing integration times, and show that here a geometrically ergodic chain can be constructed when tails are heavier than Exponential.
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Kadhem, Safaa K. "Model fit diagnostics for hidden Markov models." Thesis, University of Plymouth, 2017. http://hdl.handle.net/10026.1/9966.
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Hidden Markov models (HMMs) are an efficient tool to describe and model the underlying behaviour of many phenomena. HMMs assume that the observed data are generated independently from a parametric distribution, conditional on an unobserved process that satisfies the Markov property. The model selection or determining the number of hidden states for these models is an important issue which represents the main interest of this thesis. Applying likelihood-based criteria for HMMs is a challenging task as the likelihood function of these models is not available in a closed form. Using the data augmentation approach, we derive two forms of the likelihood function of a HMM in closed form, namely the observed and the conditional likelihoods. Subsequently, we develop several modified versions of the Akaike information criterion (AIC) and Bayesian information criterion (BIC) approximated under the Bayesian principle. We also develop several versions for the deviance information criterion (DIC). These proposed versions are based on the type of likelihood, i.e. conditional or observed likelihood, and also on whether the hidden states are dealt with as missing data or additional parameters in the model. This latter point is referred to as the concept of focus. Finally, we consider model selection from a predictive viewpoint. To this end, we develop the so-called widely applicable information criterion (WAIC). We assess the performance of these various proposed criteria via simulation studies and real-data applications. In this thesis, we apply Poisson HMMs to model the spatial dependence analysis in count data via an application to traffic safety crashes for three highways in the UK. The ultimate interest is in identifying highway segments which have distinctly higher crash rates. Selecting an optimal number of states is an important part of the interpretation. For this purpose, we employ model selection criteria to determine the optimal number of states. We also use several goodness-of-fit checks to assess the model fitted to the data. We implement an MCMC algorithm and check its convergence. We examine the sensitivity of the results to the prior specification, a potential problem given small sample sizes. The Poisson HMMs adopted can provide a different model for analysing spatial dependence on networks. It is possible to identify segments with a higher posterior probability of classification in a high risk state, a task that could prioritise management action.
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Houssineau, Jérémie. "Representation and estimation of stochastic populations." Thesis, Heriot-Watt University, 2015. http://hdl.handle.net/10399/3223.
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This work is concerned with the representation and the estimation of populations composed of an uncertain and varying number of individuals which can randomly evolve in time. The existing solutions that address this type of problems make the assumption that all or none of the individuals are distinguishable. In other words, the focus is either on specific individuals or on the population as a whole. Theses approaches have complimentary advantages and drawbacks and the main objective in this work is to introduce a suitable representation for partially-indistinguishable populations. In order to fulfil this objective, a sufficiently versatile way of quantifying different types of uncertainties has to be studied. It is demonstrated that this can be achieved within a measure-theoretic Bayesian paradigm. The proposed representation of stochastic populations is then used for the introduction of various filtering algorithms from the most general to the most specific. The modelling possibilities and the accuracy of one of these filters are then demonstrated in different situations.
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Butler, Paul. "Characterisation of disordered structures." Thesis, University of Kent, 2017. https://kar.kent.ac.uk/62479/.
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In this thesis I will look at how large, complex structures can be interpreted and evaluated using an information theoretic approach. The work specifically investigates techniques to understand disordered materials. It explains a novel framework using statistical methods to investigate structural information of very large data sets. This framework facilitates understanding of complex structures through the quantification of information and disorder. Large scale structures including granular media and amorphous atomic systems can also be processed. The need to deal with larger complex structures has been driven by new methods used to characterise amorphous materials, such as atomic scale tomography. In addition, computers are allowing for the creation of larger and larger data sets for researchers to analyse, requiring new techniques for storing and understanding information. As it has become possible to analyse large complex systems there has been a corresponding increase in attempts to scientifically understand these systems. New, man-made, complex systems have emerged such as the stock market and on-line networks. This has boosted interest in their interpretation, with the hopes they can be more easily manipulated or controlled. Crystallography has been applied to great effect in biology, having been used to discover the structure of DNA and develop new drugs (UNESCO,2013). However it only describes crystal structure, which can be a drawback as a large majority of matter is amorphous. As such it is hoped that interpreting and understanding disorder may lead to similar breakthroughs in disordered materials. Entropic measures such as the mutual information and Kullback Leibler Divergence are used to investigate the nature of structural information and its impact on the system. I examine how this information propagates in a system, and how it could quantify the amount of organisation in a system that is structurally disordered. The methodology introduced in this thesis extracts useful information from large data sets to allow for a quantification of disorder. The calculated entropy for amorphous packings is generally less than 1 bit with Mutual information between 0 and 0.1 bits. The results verify direct correlation between Mutual Information and the correlation coefficient using various techniques. The Mutual information shows most information is obtained where sphere density is highest, following a similar trend to that of the Radial distribution function, and generally increasing for higher packing fractions. Evidence of the Random Close Packed (RCP) and Random Loose Packed (RLP) limits in two dimensions is shown, as well as evidence of both phases in time-lapsed 3D packings. The Kullback Leibler Divergence is also explored as a relative measure of disorder. This is achieved by calculating redundant information in packings so that areas of low and high order can be shown. Results present colour maps displaying relative information in random disk packings from which motifs can be identified. For higher packing fractions distinct borders form for areas of low and high information, particularly where crystallisation has occurred. Again, these results show an increase in information for more densely packed structures, as expected, with a Kullback Leibler divergence of between 0 and 1 bits. Finally I introduce the concept of self-referential order which provides a way to quantify structural organisation in non-crystalline materials, by referencing part of the system in a similar way to a unit cell. This allows a step forward in understanding and characterising disorder, helping to develop a framework to encode amorphous structures in an efficient way. These results show increasing information for higher packing fractions as well as further evidence of RLP and RCP limits around packing fractions of 0.54 and 0.64 respectively.
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Samci, Karadeniz Rukiye. "Modelling share prices as a random walk on a Markov chain." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/40129.
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In the financial area, a simple but also realistic means of modelling real data is very important. Several approaches are considered to model and analyse the data presented herein. We start by considering a random walk on an additive functional of a discrete time Markov chain perturbed by Gaussian noise as a model for the data as working with a continuous time model is more convenient for option prices. Therefore, we consider the renowned (and open) embedding problem for Markov chains: not every discrete time Markov chain has an underlying continuous time Markov chain. One of the main goals of this research is to analyse whether the discrete time model permits extension or embedding to the continuous time model. In addition, the volatility of share price data is estimated and analysed by the same procedure as for share price processes. This part of the research is an extensive case study on the embedding problem for financial data and its volatility. Another approach to modelling share price data is to consider a random walk on the lamplighter group. Specifically, we model data as a Markov chain with a hidden random walk on the lamplighter group Z3 and on the tensor product of groups Z2 ⊗ Z2. The lamplighter group has a specific structure where the hidden information is actually explicit. We assume that the positions of the lamplighters are known, but we do not know the status of the lamps. This is referred to as a hidden random walk on the lamplighter group. A biased random walk is constructed to fit the data. Monte Carlo simulations are used to find the best fit for smallest trace norm difference of the transition matrices for the tensor product of the original transition matrices from the (appropriately split) data. In addition, splitting data is a key method for both our first and second models. The tensor product structure comes from the split of the data. This requires us to deal with the missing data. We apply a variety of statistical techniques such as Expectation- Maximization Algorithm and Machine Learning Algorithm (C4.5). In this work we also analyse the quantum data and compute option prices for the binomial model via quantum data.
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Ho, Chinpang. "Multilevel algorithms for the optimization of structured problems." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43384.
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Although large scale optimization problems are very difficult to solve in general, problems that arise from practical applications often exhibit particular structure. In this thesis we study and improve algorithms that can efficiently solve structured problems. Three separate settings are considered. The first part concerns the topic of singularly perturbed Markov decision processes (MDPs). When a MDP is singularly perturbed, one can construct an aggregate model in which the solution is asymptotically optimal. We develop an algorithm that takes advantage of existing results to compute the solution of the original model. The proposed algorithm can compute the optimal solution with a reduction in complexity without any penalty in accuracy. In the second part, the class of empirical risk minimization (ERM) problems is studied. When using a first order method, the Lipschitz constant of the empirical risk plays a crucial role in the convergence analysis and stepsize strategy of these problems. We derive the probabilistic bounds for such Lipschitz constants using random matrix theory. Our results are used to derive the probabilistic complexity and develop a new stepsize strategy for first order methods. The proposed stepsize strategy, Probabilistic Upper-bound Guided stepsize strategy (PUG), has a strong theoretical guarantee on its performance compared to the standard stepsize strategy. In the third part, we extend the existing results on multilevel methods for unconstrained convex optimization. We study a special case where the hierarchy of models is created by approximating first and second order information of the exact model. This is known as Galerkin approximation, and we named the corresponding algorithm Galerkin-based Algebraic Multilevel Algorithm (GAMA). Three case studies are conducted to show how the structure of a problem could affect the convergence of GAMA.
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Wu, Ruhao. "Gaussian process and functional data methods for mortality modelling." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/39143.
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Modelling the demographic mortality trends is of great importance due to its considerable impact on welfare policy, resource allocation and government planning. In this thesis, we propose to use various statistical methods, including Gaussian process (GP), principal curve, multilevel functional principal component analysis (MFPCA) for forecasting and clustering of human mortality data. This thesis is actually composed of three main topics regarding mortality modelling. In the first topic, we propose a new Gaussian process regression method and apply it to the modelling and forecasting of age-specific human mortality rates for a single population. The proposed method incorporates a weighted mean function and the spectral mixture covariance function, hence provides better performance in forecasting long term mortality rates, compared with the conventional GPR methods. The performance of the proposed method is also compared with Lee-Miller model and the functional data model by Hyndman and Ullah (2007) in the context of forecasting the French total mortality rates. Then, in the second topic, we extend mortality modelling for a single population independently to that for multiple populations simultaneously, by developing a new framework for coherent modelling and forecasting of mortality rates for multiple subpopulations within one large population. We treat the mortality of subpopulations as multilevel functional data and then a weighted multilevel functional principal component approach is proposed and used for modelling and forecasting the mortality rates. The proposed model is applied to sex-specific data for nine developed countries, and the forecasting results suggest that, in terms of overall accuracy, the model outperforms the independent model (Hyndman and Ullah 2007) and is comparable to the Product-Ratio model (Hyndman et al 2013) but with several advantages. Finally, in the third topic, we introduce a clustering method based on principal curves for clustering of human mortality as functional data. And this innovative clustering method is applied to French total mortality data for exploring its potential features.
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Kollias-Liapis, Spyridon. "Stochastic control under partial information." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43756.
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In this thesis, we consider the problem of continuous-time stochastic control with full and partial information and quadratic costs. Under some assumptions we reduce the problem of controlling a general diffusion process into controlling a piecewise linear system, called the Linearized system. The Linearized system is defined with respect to a time-partition of the fixed horizon [0,T]. We initially prove that the cost functional associated with the Linearized system converges to the cost functional of the original system as the mesh of the partition goes to 0. This in turn implies that an optimal control for the approximating system is also ε-optimal for the original system. Hence we centre our analysis at obtaining the optimal control for the Linearized system. To this end, we present two methodologies : the Perturbation method and the Policy Improvement method. In the first method, by imposing boundedness assumptions on the coefficients of the controlled diffusion, we construct the optimal control in each subinterval of the partition based on the framework of the so-called Linear Quadratic Regulator problem. In the second method we construct the optimal control in each subinterval of the partition by using a criterion under which, by starting from an arbitrary control and an associated cost, we eventually obtain, after consecutive steps, the control which minimises the cost functional of the Linearized system.
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Blacque-Florentin, Pierre. "Some infinite dimensional topics in probability and statistics." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43537.
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This thesis comprises two independent parts. In the first part, we develop a pathwise calculus for functionals of integer-valued measures and extend the framework of Functional Itô Calculus to functionals of integer-valued random measures by constructing a ’stochastic derivative’ operator with respect to such integer-valued random measures. This allows us to obtain weak martingale representation formulae holding beyond the class of Poisson random measures, and allowing for random and time-dependent compensators. We study the behaviour of this operator and compare it with other previous approaches in the literature, providing in passing a review of the various Malliavin approaches for jump processes. Finally, some examples of computations are provided. The second part is oriented towards nonparametric statistics, with a financial application as our main goal: we aim at recovering a surface of FX call options on a pegged currency such as the Hong Kong dollar against the U.S. dollar, based on a small number of noisy measurements (the market bid-ask quotes). Inspiring ourselves from the Compressed Sensing literature, we develop a methodology that aims at recovering an arbitrage-free call surface. We first apply this methodology, based on tensor polynomial decomposition of the surface, to a sparse set of call-option prices on the S&P500, recovering the call option prices within desired tolerance, as well as a smooth local-volatility surface. On a pegged currency such as the HKD/USD, it appears that tensor polynomials may not be an adequate way to model the smiles across maturities. Modifying the methodology in favour of structure-preserving functions, we apply the new methodology to our HKD/USD dataset, recovering the smiles, and the corresponding state-price density.
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Zak, Frantisek. "Long time behaviour of infinite dimensional stochastic processes." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/44084.
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We study two examples of infinite dimensional stochastic processes. Situations and techniques involved are quite varied, however in both cases we achieve a progress in describing their long time behaviour. The first case concerns interacting particle system of diffusions. We construct rigorously the process using finite dimensional approximation and the notion of martingale solution. The existence of invariant measure for the process is proved. The novelty of the results lies in the fact, that our methods enable us to consider such examples, where the generator of the diffusion is subelliptic. The other project is related to stochastic partial differential equations and their stability properties. In particular it is shown that Robbins-Monro procedure can be extended to infinite dimensional setting. Thus we achieve results about pathwise convergence of solution. To be able to define corresponding solution, we rely on so-called variational approach to stochastic partial differential equations as pioneered by E. Pardoux, N. Krylov and B. Rozovskii. Our examples covers situations such as p-Laplace operator or Porous medium operator.
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Vangelov, Borislav. "Unravelling biological processes using graph theoretical algorithms and probabilistic models." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/44521.
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This thesis develops computational methods that can provide insights into the behaviour of biomolecular processes. The methods extract a simplified representation/model from samples characterising the profiles of different biomolecular functional units. The simplified representation helps us gain a better understanding of the relations between the functional units or between the samples. The proposed computational methods integrate graph theoretical algorithms and probabilistic models. Firstly, we were interested in finding proteins that have a similar role in the transcription cycle. We performed a clustering analysis on an experimental dataset using a graph partitioning algorithm. We found groups of proteins associated with different stages of the transcription cycle. Furthermore, we estimated a network model describing the relations between the clusters and identified proteins that are representative for a cluster or for the relation between two clusters. Secondly, we proposed a computational framework that unravels the structure of a biological process from high-dimensional samples characterising different stages of the process. The framework integrates a feature selection procedure and a feature extraction algorithm in order to extract a low-dimensional projection of the high-dimensional samples. We analysed two microarray datasets characterising different cell types part of the blood system and found that the extracted representations capture the structure of the hematopoietic stem cell differentiation process. Furthermore, we showed that the low-dimensional projections can be used as a basis for analysis of gene expression patterns. Finally, we introduced the geometric hidden Markov model (GHMM), a probabilistic model for multivariate time series data. The GHMM assumes that the time series lie on a noisy low-dimensional manifold and infers a dynamical model that reflects the low-dimensional geometry. We analysed multivariate time series data generated with a stochastic model of a biomolecular circuit and showed that the estimated GHMM captures the oscillatory behaviour of the circuit.
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Wang, Jian. "Real time estimation of multivariate stochastic volatility models." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/16786/.
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This thesis firstly considers a modelling framework for multivariate volatility in financial time series. As most financial returns exhibit heavy tails and skewness, we are considering a model for the returns based on the skew-t distribution, while the volatility is assumed to follow a Wishart autoregressive process. We define a new type of Wishart autoregressive process and highlight some of its properties and some of its advantages. Particle filter based inference for this model is discussed and a novel approach of estimating static parameters is provided. Furthermore, an alternative methodology for estimating higher dimension data is developed. Secondly, inspired from the idea of Ulig's Wishart process, a new Wishart-Newton model is developed. The approach combines conjugate Bayesian inference while the hyper parameters are estimated by a Newton-Raphson method and here an online volatility estimate algorithm is proposed. The two proposed models are compared with the benchmarking GO-GARCH model in both function execution time and cumulative returns of different dimensional datasets.
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Hua, H. "Optimal and robust control for a class of nonlinear stochastic systems." Thesis, University of Liverpool, 2016. http://livrepository.liverpool.ac.uk/3001023/.
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This thesis focuses on theoretical research of optimal and robust control theory for a class of nonlinear stochastic systems. The nonlinearities that appear in the diffusion terms are of a square-root type. Under such systems the following problems are investigated: optimal stochastic control in both finite and infinite horizon; robust stabilization and robust H∞ control; H₂/H∞ control in both finite and infinite horizon; and risk-sensitive control. The importance of this work is that explicit optimal linear controls are obtained, which is a very rare case in the nonlinear system. This is regarded as an advantage because with explicit solutions, our work becomes easier to be applied into the real problems. Apart from the mathematical results obtained, we have also introduced some applications to finance.
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Coca, Cabrero Alberto Jesús. "Efficient nonparametric inference for discretely observed compound Poisson processes." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/263221.
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Compound Poisson processes are the textbook example of pure jump stochastic processes and the building blocks of Lévy processes. They have three defining parameters: the distribution of the jumps, the intensity driving the frequency at which these occur, and the drift. They are used in numerous applications and, hence, statistical inference on them is of great interest. In particular, nonparametric estimation is increasingly popular for its generality and reduction of misspecification issues. In many applications, the underlying process is not observed directly but at discrete times. Therefore, important information is missed between observations and we face a (non-linear) inverse problem. Using the intimate relationship between Lévy processes and infinite divisible distributions, we construct new estimators of the jump distribution and of the so-called Lévy distribution. Under mild assumptions, we prove Donsker theorems for both (i.e. functional central limit theorems with the uniform norm) and identify the limiting Gaussian processes. This allows us to conclude that our estimators are efficient, or optimal from an information theory point of view, and to give new insight into the topic of efficiency in this and related problems. We allow the jump distribution to potentially have a discrete component and include a novel way of estimating the mass function using a kernel estimator. We also construct new estimators of the intensity and of the drift, and show joint asymptotic normality of all the estimators. Many relevant inference procedures are derived, including confidence regions, goodness-of-fit tests, two-sample tests and tests for the presence of discrete and absolutely continuous jump components. In related literature, two apparently different approaches have been taken: a natural direct approach, and the spectral approach we use. We show that these are formally equivalent and that the existing estimators are very close relatives of each other. However, those from the first approach can only be used in small compact intervals in the positive real line whilst ours work on the whole real line and, furthermore, are the first to be efficient. We describe how the former can attain efficiency and propose several open problems not yet identified in the field. We also include an exhaustive simulation study of our and other estimators in which we illustrate their behaviour in a number of realistic situations and their suitability for each of them. This type of study cannot be found in existing literature and provides several insights not yet pointed out and solid understanding of the practical side of the problem on which real-data studies can be based. The implementation of all the estimators is discussed in detail and practical recommendations are given.
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Chevyrev, Ilya. "Characteristic functions of path signatures and applications." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:3dd5e063-bde0-434f-a781-61d3fe22aaa1.
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The main object of study in this work is the extension of the classical characteristic function to the setting of path signatures. Our first fundamental result exhibits the following geometric interpretation: the path signature is completely determined by the development of the path into compact Lie groups. This faithful representation of the signature is the primary tool we use to define and study the characteristic function. Our investigation of the characteristic function can be divided into two parts. First, we employ the characteristic function to study the expected signature of a path as the natural generalisation of the moments of a real random variable. In this direction, we provide a solution to the moment problem, and study analyticity properties of the characteristic function. In particular, we solve the moment problem for signatures arising from families of Gaussian and Markovian rough paths. Second, we study the characteristic function in relation to the solution map of a rough differential equation. The connection stems from the fact that the signature of a geometric rough path completely determines the path's role as a driving signal. As an application, we demonstrate that the characteristic function can be used to determine weak convergence of flows arising from rough differential equations. Along the way, we develop tools to study càdlàg processes as rough paths and to determine tightness in p-variation topologies of random walks. As a consequence, we provide a classification of Lévy processes possessing sample paths of finite p-variation and determine a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process.