This dissertation examines properties and representations of several isotropic Gaussian random fields in the unit ball in d-dimensional Euclidean space. First we consider Lévy's Brownian motion. We use an integral representation for the covariance function to find a new expansion for Lévy's Brownian motion as an infinite linear combination of independent standard Gaussian random variables and orthogonal polynomials. Next we introduce a new family of isotropic Gaussian random fields, called the p-processes, of which Lévy's Brownian motion is a special case. Except for Lévy's Brownian motion the p-processes are not locally stationary. All p-processes also have a representation as an infinite linear combination of independent standard Gaussian random variables. We use these expansions of the random fields to simulate Lévy's Brownian motion and the p-processes along a ray from the origin using the Cholesky factorization of the covariance matrix. / Graduation date: 2013

We use the Garman-Kohlhagen model to compute the hedge and price of a participating forward contract on the US dollar that is written by a Turkish Bank. The algorithm is computed using actual market data and a weekly updated hedge is computed. We note that despite a weekly update and many assumptions made on the volatility and the interest rates the model gives a very reasonable hedge.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

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.

Colloidal particles are ideal models for studying the behavior of atomic and molecular systems. They resemble their atomic and molecular analogues in that their dynamics are driven by thermal energy and their equilibrium properties are controlled by inter-particle interactions. Based on this analogy, it is reasonable to construct colloidal chains, where each particle represents a repeat unit, as models for polymers. The advantages of this system over molecular systems are its controllable rigidity, contour length and diameter, as well as the convenience to capture its instantaneous shape and position via video microscopy, which are not trivial to realize in molecular systems. By utilizing the dipolar properties of magnetic colloids, a number of groups have assembled semiflexible and rigid colloidal chains by cross-linking magnetic beads under a magnetic field using polymer linkers. Recently, efforts in constructing colloidal chains led even to anisotropic magnetic colloidal chains that mimic the detailed atomic arrangements of polymers. These properties make colloidal chains possible candidates for the classic bead-spring or bead-rod model systems for semiflexible and rigid polymers. In my thesis, I present a method for generating linear colloidal chain structures by linking surface functionalized paramagnetic particles using DNA. First, I investigate the force interactions between individual magnetic particles under different conditions to optimize the resulting chain stability. A systematic study the bending and rotational diffusion dynamics of the chains and their relationship with the DNA linking chemistry is presented. I then demonstrate their use as a ideal model system to study polymer dynamics In addition, a technique to measure short-range repulsive surface forces between these colloids with high precision was developed. Building on these repulsive force studies, a colloidal system to study 2-D phase transitions was created. This thesis provides insights into understanding and engineering the directed-assembly of magnetic colloids with specific surface interactions, as well as using the assemblies as model systems to study molecular level phenomena.

Modelling defaultable contingent claims has attracted a lot of interest in recent years, motivated in particular by the Late-2000s Financial Crisis. In several papers various approaches on the subject have been made. This thesis tries to summarize these results and derive explicit formulas for the prices of financial derivatives with credit risk. It is divided into two main parts. The first one is devoted to the well-known theory of modelling the default risk while the second one presents the results concerning pricing of the defaultable models that we obtained ourselves.

In this paper, we focus on the goodness of fit test for self-similar property of two well-known processes: the fractional Brownian motion and the fractional autoregressive integrated moving average process. The Hurst parameter of the self-similar process is estimated by the embedding branching process method proposed by Jones and Shen (2004). The goodness of fit test for self-similarity is based on the Pearson chi-square test statistic. We approximate the null distribution of the test statistic by a scaled chi-square distribution to correct the size bias problem of the conventional chi-square distribution. The scale parameter and degrees of freedom of the test statistic are determined via regression method. Simulations are performed to show the finite sample size and power of the proposed test. Empirical applications are conducted for the high frequency financial data and human heart rate data.

The Black-Scholes formula was the widely-used model for option pricing, this formula can be use to calculate the price of option by using current underlying asset prices, strike price, expiration time, volatility and interest rates. The European call option price from the model is a convex and increasing with respect to the initial underlying asset price. Assume underlying asset prices follow a generalized geometric Brownian motion, it is true that option prices increasing with respect to the constant interest rate and volatility, so that the volatility can be a very important factor in pricing option, if the volatility process £m(t) is constant (with £m(t) =£m for any t ) satisfying £m_1 ≤ £m(t) ≤ £m_2 for some constants £m_1 and £m_2 such that 0 ≤ £m_1 ≤ £m_2. Let C_i(t, S_t) be the price of the call at time t corresponding to the constant volatility £m_i (i = 1,2), we will derive that the price of call option at time 0 in the model with varying volatility belongs to the interval [C_1(0, S_0),C_2(0, S_0)].

In part I, MgO and Co1-xO powders in 9:1 and 1:9 molar ratio (denoted as M9C1 and M1C9 respectively) were sintered and homogenized at 1600oC followed by annealing at 850 and 800oC, respectively to form defect clusters and precipitates. Analytical electron microscopic (AEM) observations indicated the protoxide remained as rock salt structure with complicated planar diffraction contrast for M9C1 sample, however with spinel paracrystal precipitated from the M1C9 sample due to the assembly of charge- and volume-compensating defects of the 4:1 type, i.e. four octahedral vacant sites surrounding one Co3+-filled tetrahedral interstitial site. The spacing of such defect clusters is 4.5 times the lattice spacing of the average spinel structure of Mg-doped Co3-dO4, indicating a higher defect cluster concentration than undoped Co3-dO4. The {111} faulting of Mg-doped Co3-dO4/Co1-xO in the annealed M1C9 sample implies the possible presence of zinc blend-type defect clusters with cation vacancies assembled along oxygen close packed (111) plane. In part II, the Mg2TiO4/MgO composites prepared by reactive sintering MgO and TiO2 powders (9:1 molar ratio) at 1600oC and then air-cooled or further aged at 900oC were studied by X-ray diffraction and (AEM) in order to characterize the microstructures and formation mechanism of nanosized Mg2TiO4 spinel precipitated from Ti-doped MgO. Expulsion of Ti4+ during cooling caused the formation of (001)-specific G.P. zone under the influence of thermal/sintering stress and then the spinel precipitates, which were about 30 nm in size and nearly spherical with {111} and {100} facets to minimize coherency strain energy and surface energy. Secondary nano-size spinel was precipitated and became site saturated during aging at 900oC, leaving a precipitate free zone at the grain boundaries of Ti-doped MgO. The intergranular spinel became progressively Ti-richer upon aging 900oC and showed <110>-specific diffuse scatter intensity likely due to short range ordering and/or onset decomposition. In part III, the Co1-xO/Co2TiO4 composite prepared by reactive sintering CoO and TiO2 powders (9:1 molar ratio) at 1450oC and then air-cooled were studied by X-ray diffraction and AEM in order to characterize the microstructures and formation mechanism of nanosized Co2TiO4 spinel precipitated from Ti-doped Co1-xO. Slight expulsion of Ti4+ during cooling caused the precipitation of nanosize Co2TiO4 spinel. Bulk site saturation also caused impingement of the Co2TiO4 precipitates upon growth. The Co3-dO4 spinel, as an oxidatin product of Co1-xO, was found to form at free surface and the Co1-xO/Co2TiO4 interface. The Co2TiO4 spinel particles formed by reactive sintering rather than precipitation were able to detach from the Co1-xO grain boundaries to reach parallel epitaxial orientation with respect to the host Co1-xO grains via Brownian-type rotation of the embedded particles. In part IV, AEM was used to study the defect microstructures of Zr-dissolved TiO2 prepared via reactive sintering the ZrO2 and TiO2 powders (8:92 in molar ratio, designated as Z8T92) at 1600oC for 24 h and then aged at 900oC for 2-200 h in air. The Zr-dissolved TiO2 with rutile structure showed dislocation arrays, defect clusters, G.P. zone, superlattice, nanometer-size domains incommensurate and commensurate superstructure, may be the precursor of ZrTi2O6 precipitates at 900oC. The rutile showed diffuse diffractions along [001] direction as a result of Zr4+ substitution for Ti4+ with volume compensating defect clusters. Incommensurate and commensurate structures, as indicated by diffraction splitting and extra diffraction along <100> and <010> directions may be attributed to the ordering and clustering process of Zr and Ti atoms in these directions. Part V, deals with the reactive sintering of ZrO2 and TiO2 powders (1:4 molar ratio) at 1400 to 1600oC in air to form orthorhombic ZrTiO4 (a-PbO2-type structure, denoted as a) and to study its epitaxial reorientation in the matrix of tetragonal TiO2 (rutile) grains with Zr4+ (15 mol %) dissolution. The epitaxial relationship of intragranular ZrTiO4 and Zr-dissolved rutile (denoted as r) was determined by electron diffraction as [010]a//[011]r; (001)a // (011)r (i.e. [100]a // [100]r; (001)a // (011)r). The reorientation of the intragranular particles in the composites can be reasonably explained by rotation of the nonepitaxial particles above a critical temperature (T/Tm > 0.8) and below a critical particle size for anchorage release at interface with respect to the host grain. Reactive sintering facilitated the reoreientation process for the particles about to detach from the grain boundaries. The Brownian rotation of the confined ZrTiO4 particles in rutile grains was activated by a beneficial lower interfacial energy for the epitaxial relationship, typically forming lath-like ZrTiO4 with (101)a/(211)r habit plane having fair match of oxygen atoms at the interface. Further aging at 900oC for 50 h in air caused modulated and periodic antiphase domains in ZrTiO4 matrix, as likely precursor of equilibrium ZrTi2O6.

Traditional financial modeling is based on semimartingale processes with stationary and independent increments. However, empirical investigations on financial data does not always support these assumptions. This contradiction showed that there is a need for new stochastic models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation between increments of a fBm changes according to its self-similarity parameter H. This property of fBm helps to capture the correlation dynamics of the data and consequently obtain better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and classical Ito formula does not exist in that case. This gives rise to need for using the white noise theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this thesis, the representation of fBm and its fundamental properties are examined. Construction of Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula and Girsanov type theorems are stated. The financial applications of fBm are mentioned and the Black&amp / Scholes price of a European call option on an asset which is assumed to follow a geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black&amp / Scholes prices of a European call option with different maturities. Afterwards, these values are compared to Black&amp / Scholes price of the same option to demonstrate the effect of long-range dependence on the option prices. Also, estimations of H at different time scales are obtained to investigate the multiscaling in financial data. An outlook of the future work is given.