On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and exponentially fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyze and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and \pi. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation.Comment: Revised versio

Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar mean-variance frontier, and which may be implemented using quadratic programming

Extensions of Stein's lemma for the skew-normal distribution

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973, 1981), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu (1994) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegel's (1993) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Stein's lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegel's formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Stein's lemma is that all rational investors will select a portfolio which lies on Markowitz's mean-variance efficient frontier. A consequence of the extension to Stein's lemma is that under multivariate skewnormality, rational investors will select a portfolio which lies on a single meanvariance-skewness efficient hyper-surface

Exploiting skewness to build an optimal hedge fund with a currency overlay

This paper documents an investigation into the use of portfolio selection methods to construct a hedge fund with a currency overlay. The fund, which is based on number of international stock and bond market indices and is constructed from the perspective of a Sterling investor, allows the individual exposures in the currency overlay to be optimally determined. As well as using traditional mean variance, the paper constructs the hedge funds using portfolio selection methods that incorporate skewness in the optimisation process. These methods are based on the multivariate skewnormal distribution, which motivates the use of a linear skewness shock. An extension to Stein's lemma gives the ability to explore the mean-variance-skewness efficient surface without the necessity to be concerned with the precise form of an individual investor's utility function. The results suggest that it is possible to use mean variance optimisation methods to build a hedge fund based on the assets and return forecasts described. The results also suggest that the inclusion of a skewness component in the optimisation is beneficial. In many of the cases reported, the skewness term contributes to an improvement in performance over and above that given by mean variance methods

The coloured world of Alain Robbe-Grillet : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in French at Massey University

Some French language throughout.This thesis is the result of a close examination of the functions of colour in the works of Alain Robbe-Grillet, undertaken in the expectation that the careful study of this limited element would reveal the finer details of some of the important characteristics of his novels and films. The Nouveau Roman and the works of Robbe-Grillet originate in a desire to produce creative literary forms which are a better representation of man's situation in the modern world of disorder and uncertainty, than the narrative forms of the traditional nineteenth century novel. An integral part of this search for new forms is the deliberate designation and subversion of the traditional conventions which Robbe-Grillet wishes to expose as neither natural nor necessary. Thus many of the colour terms in his works are used in ironic games with these traditional forms. The illusion of realism is ironically subverted by colour and lighting references, which "foreground" the text as a fabrication of words, and also reveal that perception of reality is a subjective function and then of only one among many "realities" possible. His works therefore constitute their own reality, without necessary reference to any world "out there". However they are "realist" in that they are constrained by the laws of physical nature, e.g. description is elaborated only with illumination. Traditional colour symbols are degraded by colour, as is the convention of character, as Robbe-Grillet shows that situation and clothing do not necessarily define character or function. Ficticious characters are not "real" people but constructions of the text. The traditional anthropomorphic relationship between man and the world is thus destroyed. A related convention subverted is "le petit détail qui fait vrai", which false colour details show to be largely meaningless. Robbe-Grillet's other important subversive use for colour is to reveal the limitations of our linguistic structures; our ability to perceive colour is not matched by our ability to describe it. Colour thus plays a significant subversive role in Robbe-Grillet's works. However, to replace the traditional narrative forms, Robbe-Grillet uses colour constructively in several ways, it becomes dynamic rather than descriptive. Colour terms, at both the level of the signifier and signified, are manipulated in games with meaning to construct new texts. Traditional colour symbols are replaced with colours which become "symbolic" only in the context of a particular novel, as each now constitutes its own reality. Changing colours show the shifting focus of a narrative and create the personality of a character, while colour oppositions give movement and rhythm to texts. Specific colours generate texts through their metaphorical associations, and metaphor itself, after initial rejection, becomes a dynamic element. Colour produces many constructive forms to replace those of the traditional novel, to thus create a new "écriture romanesque". The obvious dual subversive-constructive function of colour indicates a constant tension within Robbe-Grillet's works, a tension which is perhaps the conflict basic to all literature. The many different functions of colour suggest that Robbe-Grillet's works contain an inherent multiplicity, functioning on several levels of meaning. And the changing functions of colour through the various works point to a continual evolution in Robbe-Grillet's creative production. Thus the Nouveau Roman of Robbe-Grillet is created through multiplicity, tension and evolution

On the numerical stability of Fourier extensions

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation. In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte, Trefethen & Kuijlaars states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner