GARCH models have been widely used in finance to model volatility ever since the introduction of the ARCH model and its extension to the generalized ARCH (GARCH) model. Lately, there has been growing interest in modelling seasonal volatility, most recently with the introduction of the multiplicative seasonal GARCH models. As an application of the multiplicative seasonal GARCH model with real data, call prices from the major stock market index of India are calculated using estimated parameter values. It is shown that a multiplicative seasonal GARCH option pricing model outperforms the Black-Scholes formula and a GARCH(1,1) option pricing formula. A parametric bootstrap procedure is also employed to obtain an interval approximation of the call price. Narrower confidence intervals are obtained using the multiplicative seasonal GARCH model than the intervals provided by the GARCH(1,1) model for data that exhibits multiplicative seasonal GARCH volatility.

Present day water spray based dust removal technologies do not effectively remove respirable submicron coal and silica dust particles in the underground coal mines causing Coal worker’s pneumoconiosis (CWP). The objective of this research was to study the electrostatic charges present in the airborne coal dust in order to develop efficient water spraying based dust removal technology where water drops charged using ionic compounds and surfactants would be used to capture the oppositely charged coal particles. In an experimental scrubbing column, coal particles dispersed in an air stream by a Fluidized Bed Aerosol Generator were captured by water drops sprayed by an atomizer. Characterization studies performed using an Aerodynamic Particle Sizer and Aerosol Electrometer showed that airborne coal particles have a significant amount of positive charge with an average of 140 elementary units of charge. The capture efficiencies of the water drops evaluated were found to be higher than those predicted by previously determined mathematical models. It was predicted that apart from the effects of Brownian diffusion, interception and impaction, the effect of Coulombic attraction was present and the charge of the water drops was predicted to be between - 2 x 10-6 C and -2 x 10-4 C.

The physical understanding of biological processes such as transcription requires the knowledge of double-stranded DNA (dsDNA) physics. A notable thermo- dynamic property of dsDNA is its denaturation, at the melting temperature, in which it unwinds into two single-stranded DNAs via the formation of denat- uration bubbles (segment of consecutive unpaired base-pairs). The dynamics of denaturation has been studied so far at the base-pair (bp) scale, ignoring conformational chain degrees of freedom. These studies do not explain the very long closure times of 20 to 100 s, measured by Altan-Bonnet et al., of 18 bps long bubbles at room temperature. In this thesis, we study the closure of pre-equilibrated large bubbles, by using Brownian dynamics simulations of two simple DNA coarse- grained models. We show that the closure occurs via two steps: rst, a fast zipping of the initial bubble occurs until a meta-stable state is reached, due to the large bending and twisting energies stored in the bubble. Then, the meta-stable bubble closes either via rotational di usion of the sti side arms until their alignment, or bubble di usion until it reaches the chain end, or locally by thermal activation, depending on the DNA length and elastic moduli. We show that the physical mechanism behind these long timescales is therefore the dynamical coupling between base-pair and chain degrees of freedom.

We consider p independent Brownian motions in ℝd. We assume that p ≥ 2 and p(d- 2) < d. Let ℓt denote the intersection measure of the p paths by time t, i.e., the random measure on ℝd that assigns to any measurable set A ⊂ ℝd the amount of intersection local time of the motions spent in A by time t. Earlier results of Chen derived the logarithmic asymptotics of the upper tails of the total mass ℓt(ℝd) as t →∞. In this paper, we derive a large-deviation principle for the normalised intersection measure t-pℓt on the set of positive measures on some open bounded set B ⊂ ℝd as t →∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the p motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set U ⊂ B. This extends earlier studies on the intersection measure by König and Mörters.

The aim of this PhD Thesis was to build and develop a stochastic calculus (in particular a stochastic integral) with respect to multifractional Brownian motion (mBm). Since the choice of the theory and the tools to use was not fixed a priori, we chose the White Noise theory which generalizes, in the case of fractional Brownian motion (fBm) , the Malliavin calculus. The first chapter of this thesis presents several notions we will use in the sequel.In the second chapter we present a construction as well as the main properties of stochastic integral with respect to harmonizable mBm.We also give Ito formulas and a Tanaka formula with respect to this mBm. In the third chapter we give a new definition, simplier and generalier of multifractional Brownian motion. We then show that mBm appears naturally as a limit of a sequence of fractional Brownian motions of different Hurst index.We then use this idea to build an integral with respect to mBm as a limit of sum of integrals with respect ot fBm. This being done we particularize this definition to the case of Malliavin calculus and White Noise theory. In this last case we compare the integral hence defined to the one we got in chapter 2. The fourth and last chapter propose a multifractional stochastic volatility model where the process of volatility is driven by a mBm. The interest lies in the fact that we can hence take into account, in the same time, the long range dependence of increments of volatility process and the fact that regularity vary along the time.Using the functional quantization theory in order to, among other things, approximate the solution of stochastic differential equations, we can compute the price of forward start options and then get and plot the implied volatility nappe that we graphically represent.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.

GARCH models have been widely used in finance to model volatility ever since the introduction of the ARCH model and its extension to the generalized ARCH (GARCH) model. Lately, there has been growing interest in modelling seasonal volatility, most recently with the introduction of the multiplicative seasonal GARCH models. As an application of the multiplicative seasonal GARCH model with real data, call prices from the major stock market index of India are calculated using estimated parameter values. It is shown that a multiplicative seasonal GARCH option pricing model outperforms the Black-Scholes formula and a GARCH(1,1) option pricing formula. A parametric bootstrap procedure is also employed to obtain an interval approximation of the call price. Narrower confidence intervals are obtained using the multiplicative seasonal GARCH model than the intervals provided by the GARCH(1,1) model for data that exhibits multiplicative seasonal GARCH volatility.

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

In this thesis, both experimental studies and numerical simulations of ultracold atoms in optical potentials are presented in a collection of nine scientific papers. In particular, noise-induced transport in dissipative optical lattices and superfluid properties of Bose-Einstein condensates have been studied. Noise is usually regarded as a complication to most systems and as something that needs to be minimized. However, in a series of experiments at Umeå University, noise has been shown to play a key role for laser-cooled cesium atoms trapped in dissipative optical lattices. By using a combination of two dissipative optical lattices, where the relative spatial phase between them can be controlled, a so-called Brownian motor can be realized, where energy can be extracted from the inherent noise. In the experiment, this energy is used to control the transport of the laser-cooled atoms in real time and along pre-designed paths. This thesis also presents a way to characterize this system in terms of energy conversion efficiency and coherence of the transport, which may allow for a more straightforward comparison with other systems that rely on noise rectification. In the studies, it is also shown that the noise triggers a downward drift due to gravity, even though the optical potential should support the atoms. Further investigation of this might help to understand the underlying principles of laser cooling, as well as showing that the system might be suitable as a flexible test bed for statistical physics. In close relation to the experimental system, two numerical simulations are also presented, one in which different ways to induce asymmetries between two periodic potentials are investigated, and one in which a proposal for detecting quantum walks is explored. In the second part of the thesis, a work from the Joint Quantum Institute is presented, where a long-lived persistent current in a toroidal Bose-Einstein condensate, held in an all-optical trap, is created. The critical velocity of the superflow is measured in the presence of a tunable barrier. The system can be seen as a first realization of an elementary closed-loop atom circuit. Finally a theoretical study of the crossover between one- and two-dimensional systems is presented, in particular the transition between a two-dimensional superfluid to a one-dimensional Mott insulator is investigated. / Medelst nio vetenskapliga artiklar presenteras i denna avhandling experimentella och teoretiska studier av ultrakalla atomer fångade i optiska potentialer. Framförallt har brusinducerade transporter och supraytande egenskaper hos Bose-Einstein-kondensat studerats.     För de flesta system betraktas brus som något negativt som bör minimeras, men i en serie experiment som redovisas i denna avhandling spelar bruset istället en avgörande positiv roll. I ett system där laserkylda atomer genom växelverkan med laserstrålar fångas i två individuella optiska kristallgitter, kan atomernas kollektiva rörelse styras genom att energi utvinns ur det inneboende bruset. I denna avhandling, genom att kontrollera de optiska potentialerna i realtid, visas att atomernas kollektiva rörelse kan styras längs förutbestämda banor med en så kallade Brownska motor. I ett annat experiment mäts verkningsgraden i omvandligen mellan brus och arbete, samt koherensen i atomtransporten. En sådan karakterisering gör att systemet blir enklare att jämföra med andra system som bygger på liknande principer. I avhandlingen presenteras också en studie där det visas att det inneboende bruset i systemet, tillsammans med en svag kraft, i detta fall från gravitation, kan skapa drifter trots att de optiska potentialerna borde vara tillräckligt djupa för att atomerna ska vara fångade. Denna upptäckt kan leda till ökad grundläggande kundskap om laserkylning. Dessutom visar det att systemet kan beskrivas med modeller från statistisk fysik. I relation till det experimentella systemet i Umeå redovisas även två teoretiska studier, en för två symmetriska periodiska potentialer och deras sätt att möjliggöra inducerade drifter med olika typ av asymmetrier, samt en annan för möjligheten att genomföra och detektera kvantvandringar.     I avhandlingen presenteras också ett experimentellt arbete utfört vid Joint Quantum Institute, där en långlivad ihållande ström i ett torusformat Bose-Einstein-kondensat har skapats i en optisk fälla. Den kritiska hastigheten på strömmen har mätts i närvaron av en ställbar optisk barriär. Detta system kan ses som en första realisation av en grundläggande atomkrets. Slutligen presenteras även en teoretisk studie av övergången mellan en- och tvådimensionella system, där fasövergången mellan superytande och Mottisolation studeras.

Thesis (Ph. D.)--Florida State University, 2006. / Advisor: Kai-Sheng Song, Florida State University,College of Arts and Sciences, Dept. of Statistics; David R. Peterson, Florida State University, College of Business, Dept. of Finance. Title and description from dissertation home page (viewed Sept. 27, 2006). Document formatted into pages; contains ix, 147 pages. Includes bibliographical references.