In this thesis a recently proposed method for the efficient approximation of solutions to high-dimensional partial differential equations has been investigated. This method, known as the Proper Generalised Decomposition (PGD), seeks a separated representation of the unknown field which leads to the solution of a series of low-dimensional problems instead of a single high-dimensional problem. This effectively bypasses the computational issue known as the `curse of dimensionality'. The PGD and its recent developments are reviewed and we present results for both the Poisson and Stokes problems. Furthermore, we investigate convergence of PGD algorithms by comparing them to greedy algorithms which have previously been studied in the non-linear approximation community. We highlight that convergence of PGD algorithms is not guaranteed when a Galerkin formulation of the problem is considered. Furthermore, it is shown that stability conditions related to weakly coercive problems (such as the Stokes problem) are not guaranteed to hold when employing a PGD approximation.

Random graphs and networks are of great importance in any fields including mathematics, computer science, statistics, biology and sociology. This research aims to develop statistical theory and methods of statistical inference for random graphs in novel directions. A major strand of the research is the development of conditional goodness-of-fit tests for random graph models and for random block graph models. On the theoretical side, this entails proving a new conditional central limit theorem for a certain graph statistics, which are closely related to the number of two-stars and the number of triangles, and where the conditioning is on the number of edges in the graph. A second strand of the research is to develop composite likelihood methods for estimation of the parameters in exponential random graph models. Composite likelihood methods based on edge data have previously been widely used. A novel contribution of the thesis is the development of composite likelihood methods based on more complicated data structures. The goals of this PhD thesis also include testing the numerical performance of the novel methods in extensive simulation studies and through applications to real graphical data sets.

The thesis focuses on processes on symplectic Gelfand-Tsetlin patterns. In chapter 4, a process with dynamics inspired by the Berele correspondence [Ber86] is presented. It is proved that the shape of the pattern is a Doob h-transform of independent random walks with h given by the symplectic Schur function. This is followed by an extension to a q-weighted version. This randomised version has itself a branching structure and is related to a q-deformation of the so2n+1-Whittaker functions. In chapter 5, we present a fully randomised process. This process q-deforms a process proposed in [WW09]. In chapter 7 we prove the convergence of the q-deformation of the so2n+1-Whittaker functions to the classical so2n+1-Whittaker functions when q → 1. Finally, in chapter 8 we turn our interest to the continuous setting and construct a process on patterns which contains a positive temperature analogue of the Dyson's Brownian motion of type B∕C. The processes obtained are h-transforms of Brownian motions killed at a continuous rate that depends on their distance from the boundary of the Weyl chamber of type B∕C, with h related with the so2n+1-Whittaker functions.

In this thesis, we investigate a class of stochastic processes whose definition can be achieved by formulating a nonlinear martingale problem and subsequently proving its well-posedness. This class includes so-called nonlinear Markov processes, such as McKean-Vlasov processes and nonlinear diffusions, but also non-Markovian versions of those. Roughly speaking, these processes are characterised by the fact that the evolution of their realisations depends on a particular finite dimensional distribution of the process itself. To formalise our idea we need to specify three components: (1) a collection of delay points which determines the finite dimensional distributions to be considered in the nonlinearity; (2) a family of operators which describes the evolution of the marginal probability distributions of the process; and (3) an initial condition which characterises the process on an initial period of time defined by the collection of delay points. Given these three elements we are able to formulate rigorously a nonlinear martingale problem and investigate its well-posedness. Our main results, which can be found in Chapter 4, provide sufficient conditions to guarantee the existence of a unique solution to the nonlinear martingale problem. The proof consists of three parts: constructing an approximating sequence of “standard” stochastic processes – together with a sequence of related curves of probability measures – proving its convergence, and finally demonstrating that its limit satisfies the martingale problem. To accomplish the proof we require a decomposition akin to the one provided by Ito’s formula. The reason why the classical Ito’s formula cannot be applied is that we need a decomposition for functions depending on the process at a finite number of non-anticipating times and not just on the process at the current time. To overcome this difficulty we establish an appropriate Ito-type formula by using Skorohod integration theory. The material related to this formula can be found in Chapter 3. In addition, in Chapter 5 we prove the existence of solutions of a class of nonlinear SDEs with unbounded coefficients by using a different approach which was proposed in Kolokoltsov, 2010 and allows to investigate a class of nonlinear stochastic processes. Finally, we present two examples of nonlinear SDEs in Chapter 6. The purpose of such examples is twofold, first illustrate that the conditions for existence of solutions are sufficient but not necessary; and second to show potential applications. The idea is to propose stochastic volatility models with nonlinear dependence. In particular, we set two models via SDEs.

In this paper we prove the well-posedness for a stochastic partial differential equation (SPDE) whose solution is a probability-measure-valued process. We allow the coefficients to depend on the median or, more generally, on the γ-quantile (or some its useful extensions) of the underlying distribution. Such SPDEs arise in many applications, for example, in auction system described in [2]. The well-posedness of this SPDE does not follow by standard arguments because the γ-quantile is not a continuous function on the space of probability measures equipped with the weak convergence.

In this thesis we study low-dimensional stochastic volatility interest rate models for pricing and hedging exotic derivatives. In particular we develop a stochastic volatility Markov-functional model. In order to implement the model numerically, we further propose a general algorithm by working with basis functions and conditional moments of the driving Markov process. Motivated by a data driven study, we choose a SABR type model as a driving process. With this choice we specify a pre-model and develop an approximation to evaluate conditional moments of the SABR driver which serve as building blocks for the practical algorithm. Having discussed how to set up a stochastic volatility Markov-functional model next we study the calibration of a LIBOR based version of the model with the SABR type driving process. We consider a link between separable SABR LIBOR market models and stochastic volatility LIBOR Markov-functional models. Based on the link we propose a calibration routine to feed in SABR marginals by calibrating to the market vanilla options. Moreover we choose the parameters of the SABR driver by fitting to the market correlation structure. We compare the stochastic volatility Markov-functional model developed in the thesis with one-dimensional (non-stochastic-volatility) swap Markov-functional models in terms of pricing and hedging Bermudan type products. By doing so we investigate effects of correlation structure, implied volatility smiles and the introduction of stochastic volatility on Bermudan type products. Finally we compare Quasi-Gaussian models with Markov-functional models in terms of specification and calibration. In particular we study Quasi-Gaussian models formulated in the Markov-functional model framework to make clear the relationship between the two models.

Markov chain Monte Carlo (MCMC) is one of the most popular statistical inference methods in machine learning. Recent work shows that a significant improvement of the statistical efficiency of MCMC on complex distributions can be achieved by exploiting geometric properties of the target distribution. This is known as geometric MCMC. However, many such methods, like Riemannian manifold Hamiltonian Monte Carlo (RMHMC), are computationally challenging to scale up to high dimensional distributions. The primary goal of this thesis is to develop novel geometric MCMC methods applicable to large-scale problems. To overcome the computational bottleneck of computing second order derivatives in geometric MCMC, I propose an adaptive MCMC algorithm using an efficient approximation based on Limited memory BFGS. I also propose a simplified variant of RMHMC that is able to work effectively on larger scale than the previous methods. Finally, I address an important limitation of geometric MCMC, namely that is only available for continuous distributions. I investigate a relaxation of discrete variables to continuous variables that allows us to apply the geometric methods. This is a new direction of MCMC research which is of potential interest to many applications. The effectiveness of the proposed methods is demonstrated on a wide range of popular models, including generalised linear models, conditional random fields (CRFs), hierarchical models and Boltzmann machines.

Bibliography: leaves 146-152 / vi, 152 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1987

First chapter of my thesis reviews recent developments in the theory and practice of volatility measurement. We review the basic theoretical framework and describe the main approaches to volatility measurement in continuous time. In this literature the central parameter of interest is the integrated variance and its multivariate counterpart. We describe the measurement of these parameters under ideal circumstances and when the data are subject to measurement error, microstructure issues. We also describe some common applications of this literature. In the second chapter, we propose a new estimator of multivariate ex-post volatility that is robust to microstructure noise and asynchronous data timing. The method is based on Fourier domain techniques. The advantage of this method is that it does not require an explicit time alignment, unlike existing methods in the literature. We derive the large sample properties of our estimator under general assumptions allowing for the number of sample points for different assets to be of different order of magnitude. We show in extensive simulations that our method outperforms the time domain estimator especially when two assets are traded very asynchronously and with different liquidity. In the third chapter, we propose to model high frequency price series by a timedeformed L´evy process. The deformation function is modeled by a piecewise linear function of a physical time with a slope depending on the marks associated with intra-day transaction data. The performance of a quasi-MLE and an estimator based on a permutation-like statistic is examined in extensive simulations. We also consider estimating the deformation function nonparametrically by pulling together many time series. We show that financial returns spaced by equal elapse of estimated deformed time are homogenous. We propose an order execution strategy using the fitted deformation time

In the field of underground radioactive waste disposal, complex computer models are used to describe the flow of groundwater through rocks. An important property in this context is transmissivity, the ability of the groundwater to pass through rocks, and the transmissivity field can be represented by a stochastic model. The stochastic model is included in complex computer models which determine the travel time for radionuclides released at one point to reach another. As well as the uncertainty due to the stochastic model, there may also be uncertainties in the inputs of these models. In order to quantify the uncertainties, Monte Carlo analyses are often used. However, for computationally expensive models, it is not always possible to obtain a large enough sample to provide accurate enough uncertainty analyses. In this thesis, we present the use of Bayesian emulation methodology as an alternative to Monte Carlo in the analysis of stochastic models. The idea behind Bayesian emulation methodology is that information can be obtained from a small number of runs of the model using a small sample from the input distribution. This information can then be used to make inferences about the output of the model given any other input. The current Bayesian emulation methodology is extended to emulate two statistics of a stochastic computer model; the mean and the distribution function of the output. The mean is a simple output statistic to emulate and provides some information about how the output changes due to changes in each input. The distribution function is more complex to emulate, however it is an important statistic since it contains information about the entire distribution of the outputs. Distribution functions of radionuclide travel times have been used as part of risk analyses for underground radioactive waste disposal. The extended methodology is presented using a case study.