This volume contains five essays in the field of time series econometrics. All five discuss properties of fractionally integrated processes and models. The first essay, entitled Do Long-Memory Models have Long Memory?, demonstrates that fractional integration can enhance the memory of ARMA processes enormously. This is however not true for all combinations of diffe-rencing, autoregressive and moving average parameters. The second essay, with the title On the Effects of Imposing or Ignoring Long-Memory when Forecasting, investigates how the choice between mo-delling stationary time series as ARMA or ARFIMA processes affect the accu-racy of forecasts. The results suggest that ignoring long-memory is worse than imposing it and that the maximum likelihood estimator for the ARFIMA model is to prefer. The third essay, Power and Bias of Likelihood Based Inference in the Cointegration Model under Fractional Cointegration, investigates the performance of the usual cointegration approach when the processes are fractionally cointegrated. Under these circumstances, it is shown that the maximum likelihood estimates of the long-run relationship are severely biased. The fourth and fifth essay, entitled respectively Bootstrap Testing for Fractional Integration and Robust Testing for Fractional Integration using the Bootstrap, propose and investigate the performance of some bootstrap testing procedures for fractional integration. The results suggest that the empirical size of a bootstrap test is (almost) always close to the nominal, and that a well-designed bootstrap test is quite robust to deviations from standard assumptions. / Diss. Stockholm : Handelshögsk. [7] s., s. x-xiv, s. 1-26: sammanfattning, s. 27-111, [4] s.: 5 uppsatser

Features inside the living cell are complex and crowded; in such complex environments diffusion processes can be said to exhibit three distinct behaviours: pure or Fickian diffusion, superdiffusion and subdiffusion. Furthermore, the behaviour of biochemical processes taking place in these environments does not follow classical theory. Because of these factors, the task of modelling dynamical proceses in complex environments becomes very challenging and demanding and has received considerable attention from other researchers seeking to construct a coherent model. Here, we are interested to study the phenomenon of subdiffusion, which occurs when there is molecular crowding. The Reaction Diffusion Partial Differential Equations (RDPDEs) approach has been used traditionally to study diffusion. However, these equations have limitations due to their unsuitability for a subdiffusive setting. However, I provide models based on Fractional Reaction Diffusion Partial Differential Equations (FRDPDEs), which are able to portray intracellular diffusion in crowded environments. In particular, we will consider a class of continuous spatial models to describe concentrations of molecular species in crowded environments. In order to investigate the variability of the crowdedness, we have used the anomalous diffusion parameter $\alpha$ to mimic immobile obstacles or barriers. We particularly use the notation $D_t^{1-\alpha} f(t)$ to represent a differential operator of noninteger order. When the power exponent is $\alpha=1$, this corresponds to pure diffusion and to subdiffusion when $0<\alpha<1$. This thesis presents results from the application of fractional derivatives to the solution of systems biology problems. These results are presented in Chapters 4, 5 and 6. An introduction to each of the problems is given at the beginning of the relevant chapter. The introduction chapter discusses intracellular environments and the motivation for this study. The first main result, given in Chapter 4, focuses on formulating a variable stepsize method appropriate for the fractional derivative model, using an embedded technique~\cite{landman07,simpson07,simpson06}. We have also proved some aspects of two fractional numerical methods, namely the Fractional Euler and Fractional Trapezoidal methods. In particular, we apply a Taylor series expansion to obtain a convergence order for each method. Based on these results, the Fractional Trapezoidal has a better convergence order than the Fractional Euler. Comparisons between variable and fixed stepsizes are also tested on biological problems; the results behave as we expected. In Chapter 5, analyses are presented related to two fractional numerical methods, Explicit Fractional Trapezoidal and Implicit Fractional Trapezoidal methods. Two results, based on Fourier series, related to the stability and convergence orders for both methods have been found. The third main result of this thesis, in Chapter 6, concerns the travelling waves phenomenon modeled on crowded environments. Here, we used the FRDPDEs developed in the earlier chapters to simulate FRDPDEs coupled with cubic or quadratic reactions. The results exhibit some interesting features related to molecular mobility. Later in this chapter, we have applied our methods to a biological problem known as Hirschsprung's disease. This model was introduced by Landman~\cite{landman07}. However, that model ignores the effects of spatial crowdedness in the system. Applying our model for modelling Hirschsprung's disease allows us to establish an interesting result for the mobility of the cellular processes under crowded environmental conditions.

The cable television (CATV) industry has grown significantly since its inception in the late 1940’s. Originally, a CATV network was comprised of several homes that were connected to community antennae via a network of coaxial cables. The only signal processing done was by an analogue amplifier, and transmission only occurred in one direction (i.e. from the antennae/head-end to the subscribers). However, as CATV grew in popularity, demand for services such as pay-per-view television increased, which lead to supporting transmission in the upstream direction (i.e. from subscriber to the head-end). This greatly increased the signal processing to include frequency diplexers. CATV service providers began to expand the bandwidth of their networks in the late 90’s by switching from analogue to digital technology. In an effort to regulate the manufacturing of new digital equipment and ensure interoperability of products from different manufacturers, several cable service providers formed a non-for-profit consortium to develop a data-over-cable service interface specification (DOCSIS). The consortium, which is named CableLabs, released the first DOCSIS standard in 1997. The DOCSIS standard has been upgraded over the years to keep up with increased consumer demand for large bandwidths and faster transmission speeds, particularly in the upstream direction. The latest version of the DOCSIS standard, DOCSIS 3.1, utilizes orthogonal frequency-division multiple access (OFDMA) technology to provide upstream transmission speeds of up to 1 Gbps. As cable service providers begin the process of upgrading their upstream receivers to comply with the new DOCSIS 3.1 standard, they require a means of testing the various functions that an upstream receiver may employ. It is convenient for service providers to employ cable modem (CM) plus channel emulator to perform these tests in-house during the product development stage. Constructing the emulator in digital technology is an attractive option for testing. This thesis approaches digital emulation by developing a digital model of the CMs and upstream channel in a DOCSIS 3.1 network. The first step in building the emulator is to simulate its operations in MATLAB, specifically upstream transmission over the network. The MATLAB model is capable of simulating transmission from multiple CMs, each of which transmits using a specific “transmission mode.” The three transmission modes described in the DOCSIS 3.1 standard are included in the model. These modes are “traffic mode,” which is used during regular data transmission; “fine ranging mode,” which is used to perform fine timing and power offset corrections; and “probing” mode, which is presumably used for estimating the frequency response of the channel, but also is used to further correct the timing and power offsets. The MATLAB model is also capable of simulating the channel impairments a signal may encounter when traversing the upstream channel. Impairments that are specific to individual CMs include integer and fractional timing offsets, micro-reflections, carrier phase offset (CPO), fractional carrier frequency offset (CFO), and network gain/attenuation. Impairments common to all CMs include carrier hum modulation, AM/FM ingress noise, and additive white Gaussian noise (AWGN). It is the hope that the MATLAB scripts that make up the simulation be translated to Verilog HDL to implement the emulator on a field-programmable gate array (FPGA) in the near future. In the event that an FPGA implementation is pursued, research was conducted into designing efficient fractional delay filters (FDFs), which are essential in the simulation of micro-reflections. After performing an FPGA implementation cost analysis between various FDF designs, it was determined that a Kaiser-windowed sinc function FDF with roll-off parameter β = 3.88 was the most cost-efficient choice, requiring at total of 24 multipliers when implemented using an optimized structure.

This thesis is concerned with a particular numerical approach for solving the fractional partial differential equations (PDEs). In the last two decades, it has been observed that many practical systems are more accurately described by fractional differential equations (FDEs) rather than the traditional differential equation approaches. Consequently, it has become an important research area to study the theoretical and numerical aspects of various types of FDEs. This thesis will explore high order numerical methods for solving FDEs numerically. More precisely, spectral methods which exhibits exponential order of accuracy will be investigated. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss quadrature points. In this work, Hermite and modified rational functions are employed to serve as basis functions for solutions that decay exponentially and algebraically, respectively. The main emphasis of this thesis is to propose the spectral collocation method for FDEs posed in unbounded domains. Components of the differentiation matrix involving fractional Laplacian are derived which can then be computed recursively using the properties of confluent hypergeometric function or hypergeometric function. The first part of the thesis introduces preliminaries useful for other parts of the thesis. Review of the relevant definitions and properties of special functions such as Hermite functions, Bessel functions, hypergeometric functions, Gegenbauer polynomials, mapped Jacobi polynomials, modified rational functions are presented. Fractional Sobolev space is introduced and some lemmas on interpolation approximation in the fractional Sobolev space are also included. In the second part of the thesis, we present the spectral collocation method based on Hermite functions. Two bases are used, namely, the over-scaled Hermite function and generalized Hermite function, which are orthogonal functions on the whole line with appropriate weight functions. We will show that the fractional Laplacian of these two kinds of Hermite functions can be represented by confluent hypergeometric function. Behaviors of the condition numbers for the resulting spectral differentiation matrices with respect to the number of expansion terms are investigated. Moreover, approximation in two-dimensional space using the tensorized bases, application to multi-term problems and use of scaling to match different decay rate are also considered. Convergence analysis for generalized Hermite function are derived and numerical errors for two bases are analyzed. The third part of the thesis deals with the spectral collocation method based on modified rational functions. We first give a brief introduction for computation of the fractional Laplacian using modified rational functions, which is represented by hypergeometric functions. Then the differentiation matrix involving the fractional Laplace operator is given. Convergence analysis for modified Chebyshev rational functions and modified Legendre rational functions are derived and numerical experiments are carried out.

A definition of fractional Fourier transform as the generalization of ordinary Fourier transform is given at the beginning. Then due to optical reasons the fractional transform of a so-called chirp functions is considered in both theory and practical simulations. Because of a quadratic phase factor which is common in the definition of the transform and some optical concepts, a comparison between these concepts such as Fresnel diffraction, spherical wave, thin lens and free space propagation and the transform has been done. Finally an optical setup for performing the fractional transform is introduced.

No description available.

The standard Black-Scholes model is under the assumption of geometric Brownian motion, and the log-returns for Black-Scholes model are independent and Gaussian. However, most of the recent literature on the statistical properties of the log-returns makes this hypothesis not always consistent. One of the ongoing research topics is to nd a better nancial pricing model instead of the Black-Scholes model. In the present work, we concentrate on two typical 1-D option pricing models under the general exponential L evy processes, namely the nite moment log-stable (FMLS) model and the the Carr-Geman-Madan-Yor-eta (CGMYe) model, and we also propose a multivariate CGMYe model. Both the frameworks, and the numerical estimations and simulations are studied in this thesis. In the future work, we shall continue to study the fractional partial di erential equations (FPDEs) of the nancial models, and seek for the e cient numerical algorithms of the American pricing problems. Keywords: fractional partial di erential equation; option pricing models; exponential L evy process; approximate solution.

An estimation methodology to deal with fractionally integrated processes of Ornstein- Uhlenbeck type is proposed. The methodology is based on the continuous Whittle contrast. A simulation study is performed by driving this process with a symmetric CGMY background Lévy process.

No description available.

This thesis concerns the development of parallel algorithms to solve fractional differential equations using a numerical approach. The methodology adopted is to adapt existing numerical schemes and to develop prototype parallel programs using the MatLab Parallel Computing Toolbox (MPCT). The approach is to build on existing insights from parallel implementation of ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence of the work, new insights on the use of MPCT for prototyping are presented, alongside conclusions and algorithms for the effective implementation of parallel methods for the fractional calculus. The principal parallel approaches considered in the work include: - A Runge-Kutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme - An implementation of the Diethelm-Chern Algorithm for Fractional Differential Equations - A parallel version of the well-established Fractional Adams Method for Fractional Differential Equations - The adaptation for parallel implementation of Lubich's Fractional Multistep Method for Fractional Differential Equations An important aspect of the work is an improved understanding of the comparative diffi culty of using MPCT for obtaining fair comparisons of parallel implementation. We present details of experimental results which are not satisfactory, and we explain how the problems may be overcome to give meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls can be put in place to allow like-for-like performance comparisons between sequential and parallel programs.