Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards’ equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form. The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

Scaling digital radiographs for templating in total hip arthroplasty using conventional acetate templates independent of calibration markers

We describe a scaling method for templating digital radiographs using conventional acetate templates independent of template magnification without the need for a calibration marker. The mean magnification factor for the radiology department was determined (119.8%, range117%-123.4%). This fixed magnification factor was used to scale the radiographs by the method described. 32 femoral heads on postoperative THR radiographs were then measured and compared to the actual size. The mean absolute accuracy was within 0.5% of actual head size (range 0 to 3%) with a mean absolute difference of 0.16mm (range 0-1mm, SD 0.26mm). Intraclass Correlation Coefficient (ICC) showed excellent reliability for both inter and intraobserver measurements with ICC scores of 0.993 (95% CI 0.988-0.996) for interobserver measurements and intraobserver measurements ranging between 0.990-0.993 (95% CI 0.980-0.997).

Reduced-Size microstrip patch antenna for Bluetooth applications

A novel reduced-size microstrip rectangular patch antenna for Bluetooth operation is presented in this paper. The proposed antenna operates in the 2400 to 2484 MHz ISM Band. Although an air substrate is introduced, antenna occupies a small volume of 33.3×6.6×0.8 mm3. The gain and the impedance bandwidth of the antenna are predicted using a commercial Finite Element Method software package. The predicted results show good agreement with measured data.

A dynamic wheel-rail impact analysis of railway track under wheel flat by Finite Element Analysis

Wheel–rail interaction is one of the most important research topics in railway engineering. It involves track impact response, track vibration and track safety. Track structure failures caused by wheel–rail impact forces can lead to significant economic loss for track owners through damage to rails and to the sleepers beneath. Wheel–rail impact forces occur because of imperfections in the wheels or rails such as wheel flats, irregular wheel profiles, rail corrugations and differences in the heights of rails connected at a welded joint. A wheel flat can cause a large dynamic impact force as well as a forced vibration with a high frequency, which can cause damage to the track structure. In the present work, a three-dimensional (3-D) finite element (FE) model for the impact analysis induced by the wheel flat is developed by use of the finite element analysis (FEA) software package ANSYS and validated by another validated simulation. The effect of wheel flats on impact forces is thoroughly investigated. It is found that the presence of a wheel flat will significantly increase the dynamic impact force on both rail and sleeper. The impact force will monotonically increase with the size of wheel flats. The relationships between the impact force and the wheel flat size are explored from this finite element analysis and they are important for track engineers to improve their understanding of the design and maintenance of the track system.

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

A new mesh adaptivity algorithm that combines a posteriori error estimation with bubble-type local mesh generation (BLMG) strategy for elliptic differential equations is proposed. The size function used in the BLMG is defined on each vertex during the adaptive process based on the obtained error estimator. In order to avoid the excessive coarsening and refining in each iterative step, two factor thresholds are introduced in the size function. The advantages of the BLMG-based adaptive finite element method, compared with other known methods, are given as follows: the refining and coarsening are obtained fluently in the same framework; the local a posteriori error estimation is easy to implement through the adjacency list of the BLMG method; at all levels of refinement, the updated triangles remain very well shaped, even if the mesh size at any particular refinement level varies by several orders of magnitude. Several numerical examples with singularities for the elliptic problems, where the explicit error estimators are used, verify the efficiency of the algorithm. The analysis for the parameters introduced in the size function shows that the algorithm has good flexibility.

Atheroma: Is calcium important or not? A modelling study of stress within the atheromatous fibrous cap in relation to position and size of calcium deposits

Atheromatous plaque rupture h the cause of the majority of strokes and heart attacks in the developed world. The role of calcium deposits and their contribution to plaque vulnerability are controversial. Some studies have suggested that calcified plaque tends to be more stable whereas others have suggested the opposite. This study uses a finite element model to evaluate the effect of calcium deposits on the stress within the fibrous cap by varying their location and size. Plaque fibrous cap, lipid pool and calcification were modeled as hyperelastic, Isotropic, (nearly) incompressible materials with different properties for large deformation analysis by assigning time-dependent pressure loading on the lumen wall. The stress and strain contours were illustrated for each condition for comparison. Von Mises stress only increases up to 1.5% when varying the location of calcification in the lipid pool distant to the fibrous cap. Calcification in the fibrous cap leads to a 43% increase of Von Mises stress when compared with that in the lipid pool. An increase of 100% of calcification area leads to a 15% stress increase in the fibrous cap. Calcification in the lipid pool does not increase fibrous cap stress when it is distant to the fibrous cap, whilst large areas of calcification close to or in the fibrous cap may lead to a high stress concentration within the fibrous cap, which may cause plaque rupture. This study highlights the application of a computational model on a simulation of clinical problems, and it may provide insights into the mechanism of plaque rupture.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.