A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

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.

An advanced implicit meshless approach for fractional partial differential equation in computational mechanics

Recently, the numerical modelling and simulation for fractional partial differential equations (FPDE), which have been found with widely applications in modern engineering and sciences, are attracting increased attentions. The current dominant numerical method for modelling of FPDE is the explicit Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of time fractional diffusion equations. The discrete system of equations is obtained by using the RBF meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation for FPDE.

Multiscale analysis of protein functions and stochastic modelling of gene transcriptional regulatory networks

Genomic and proteomic analyses have attracted a great deal of interests in biological research in recent years. Many methods have been applied to discover useful information contained in the enormous databases of genomic sequences and amino acid sequences. The results of these investigations inspire further research in biological fields in return. These biological sequences, which may be considered as multiscale sequences, have some specific features which need further efforts to characterise using more refined methods. This project aims to study some of these biological challenges with multiscale analysis methods and stochastic modelling approach. The first part of the thesis aims to cluster some unknown proteins, and classify their families as well as their structural classes. A development in proteomic analysis is concerned with the determination of protein functions. The first step in this development is to classify proteins and predict their families. This motives us to study some unknown proteins from specific families, and to cluster them into families and structural classes. We select a large number of proteins from the same families or superfamilies, and link them to simulate some unknown large proteins from these families. We use multifractal analysis and the wavelet method to capture the characteristics of these linked proteins. The simulation results show that the method is valid for the classification of large proteins. The second part of the thesis aims to explore the relationship of proteins based on a layered comparison with their components. Many methods are based on homology of proteins because the resemblance at the protein sequence level normally indicates the similarity of functions and structures. However, some proteins may have similar functions with low sequential identity. We consider protein sequences at detail level to investigate the problem of comparison of proteins. The comparison is based on the empirical mode decomposition (EMD), and protein sequences are detected with the intrinsic mode functions. A measure of similarity is introduced with a new cross-correlation formula. The similarity results show that the EMD is useful for detection of functional relationships of proteins. The third part of the thesis aims to investigate the transcriptional regulatory network of yeast cell cycle via stochastic differential equations. As the investigation of genome-wide gene expressions has become a focus in genomic analysis, researchers have tried to understand the mechanisms of the yeast genome for many years. How cells control gene expressions still needs further investigation. We use a stochastic differential equation to model the expression profile of a target gene. We modify the model with a Gaussian membership function. For each target gene, a transcriptional rate is obtained, and the estimated transcriptional rate is also calculated with the information from five possible transcriptional regulators. Some regulators of these target genes are verified with the related references. With these results, we construct a transcriptional regulatory network for the genes from the yeast Saccharomyces cerevisiae. The construction of transcriptional regulatory network is useful for detecting more mechanisms of the yeast cell cycle.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

Stochastic models and simulation of ion channel dynamics

The behaviour of ion channels within cardiac and neuronal cells is intrinsically stochastic in nature. When the number of channels is small this stochastic noise is large and can have an impact on the dynamics of the system which is potentially an issue when modelling small neurons and drug block in cardiac cells. While exact methods correctly capture the stochastic dynamics of a system they are computationally expensive, restricting their inclusion into tissue level models and so approximations to exact methods are often used instead. The other issue in modelling ion channel dynamics is that the transition rates are voltage dependent, adding a level of complexity as the channel dynamics are coupled to the membrane potential. By assuming that such transition rates are constant over each time step, it is possible to derive a stochastic differential equation (SDE), in the same manner as for biochemical reaction networks, that describes the stochastic dynamics of ion channels. While such a model is more computationally efficient than exact methods we show that there are analytical problems with the resulting SDE as well as issues in using current numerical schemes to solve such an equation. We therefore make two contributions: develop a different model to describe the stochastic ion channel dynamics that analytically behaves in the correct manner and also discuss numerical methods that preserve the analytical properties of the model.

Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focused recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the "best" choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.

Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.