Collateralization in over-the-counter (OTC) derivatives markets has grown rapidly over the past decade, and even faster in the past few years, due to the impact of the recent financial crisis and the particularly important attention to the counterparty credit risk in derivatives contracts. The addition of collateralization to such contracts significantly reduces the counterparty credit risk and allows to offset liabilities in case of default. We study the problem of valuation of OTC derivatives with payoff in a single currency and with single underlying asset for the cases of zero, partial, and perfect collateralization. We assume the derivative is traded between two default-free counterparties and analyze the impact of collateralization on the fair present value of the derivative. We establish a uniform generalized derivative pricing framework for the three cases of collateralization and show how different approaches to pricing turn out to be consistent. We then generalize the results to include multi-asset and cross-currency arguments, where the underlying and the derivative are in some domestic currency, but the collateral is posted in a foreign currency. We show that the results for the single currency, multi-asset case are consistent with those obtained for the single currency, single asset case.

<p> A method for the simultaneous preconcentration by solvent extraction of a group of trace metal ions from natural waters has been developed. The procedure makes use of a proprietary "liquid cation-exchanger", Kelex 100, the primary component of which is an alkylated oxine (8-quinolinol) derivative, 7-(4-ethyl-1-methyloctyl)-8-quinolinol (HL). After purification of HL from the commercial mixture, the extraction of ten environmentally-significant trace metal ions from artificial seawater into toluene solution was studied as a function of pH. From these investigations, the optimal conditions for the extraction of Cd(II), Co(II), Cu(II), Mn(II), Ni(II), Pb(II) and Zn(II) from natural waters were established. The conditions for quantitative back-extraction of the metal ions were then investigated. With the exception of cobalt, the metal ions were quantitatively back-extracted into a small volume of nitric acid, simplifying the matrix and providing additional analyte enrichment. The optimized forward- and back-extraction technique was subsequently applied to the determination of total (soluble) Cd, Cu, Mn, Ni and Pb in a coastal seawater reference standard by graphite-furnace atomic absorption spectroscopy (GFAAS). The quantitative recovery of the analytes and the uncomplicated matrix of analysis enabled quantitation to be carried out by external calibration. Compared to the method of standard additions, external calibration has advantages in overall analysis time and sample consumption. Satisfactory agreement was obtained between the experimental and reference values, although Cu(II) blanks were high due to trace Cu(II) contamination of HL and the stability of the Cu(II)-HL chelate.</p> <p> The lipophilicity of HL and its metal chelates provided high metal chelate distribution ratios which, in turn, permitted preconcentration factors of up to 500 in a single batch-extraction. Additionally, studies on the recovery of radiotracer spikes from lakewater and seawater suggested that HL is an effective extractant for stripping metal ions from variously-bound forms from natural waters.</p> / Thesis / Master of Science (MSc)

Electromagnetism and optimal control stand out as a topics that feature impactful applications in modern engineering, as well as challenging theoretical aspects of mathematical analysis. Within this context, a major role is played by the search of necessary and sufficient conditions characterizing optimal solutions, as they are functional to numerical algorithms aiming to approximate such solutions. In this thesis, three standalone topics in optimization sharing the underlying framework of Maxwell-related PDEs are discussed. First, I present an optimal control problem driven by a quasi-linear magneto-static obstacle problem featuring first-order differential state constraints. The non-linearity allows to suitably model electromagnetic waves in the presence of ferromagnetic materials, while the first-order obstacle is relevant for applications in the field of magnetic shielding. Existence theory and the derivation of an optimality system are addressed with an approximation technique based on a relaxation-penalization of the variational inequality. Second, I analyze an eddy current problem controlled through a dipole type source, i.e. a Dirac mass with fixed position and variable intensity: well-posedness of the state equation through a fundamental solution (of a curl curl - Id operator) approach and first order conditions are dealt with. To conclude, I discuss the computation of the topological derivative for shape functionals constrained to low-frequency electromagnetic problems (closely related to the eddy current model), with respect to the inclusion/removal of conducting material; the results are obtained using a Lagrangian approach and in particular the so-called averaged adjoint method. This approach requires the study of the asymptotic behavior of the solutions of some problems defined in the whole space, and the introduction and consequent analysis of appropriate function spaces.

In this thesis, we provide a comprehensive overview of classical fractional derivatives and collect results on mapping properties. In particular, we discuss mapping properties e.g. we prove that the 𝛼 order fractional derivative maps the Sobolev space W_0^(p,s) to the fractional Sobolev-Slobodeckij space W^(p,s-α) for all 𝛼 < 𝑠 < 1. Further, we present several definitions of “Bernstein fractional derivatives” using the Bernstein function and in particular, we study the Bernstein censored fractional derivative by using the Picard method to get its inverse Bernstein censored fractional integral. Moreover, we use analytic tools to get the existence and uniqueness of the solution of the corresponding resolvent equation. Finally, we construct a stochastic process through Ikeda–Nagasawa–Watanabe (INW) piecing together procedure such that its generator is the Bernstein censored fractional derivative. Additionally, we show that this process gives a Feller semigroup.:Introduction 1 Basics 1.1 Some results in functional analysis 1.2 Fourier, Laplace and Mellin transforms 1.3 Regularly varying functions 1.4 Markov processes 1.5 Lévy processes and subordinators 2 Fractional derivatives and integrals 2.1 Classical fractional integrals and derivatives 2.2 Mapping properties of fractional integrals and derivatives 2.3 Bernstein functions 2.4 Fractional derivatives based on Bernstein Functions 2.5 Probabilistic interpretation of fractional derivatives 2.6 Fractional Laplace operator 3 Censored Bernstein fractional derivative and integral 3.1 Sonine pairs 3.2 Examples of Sonine pairs 3.3 Mapping properties of general fractional derivatives 3.4 Censored Bernstein fractional derivative and integral 3.5 Linear censored initial value problem 4 Censored process 4.1 Construction 4.2 Probabilistic representation 5 Application 5.1 Censored subordinator for a regularly varying kernel 5.2 Linear censored initial value problem for regularly varying kernels Bibliography

In calculus the concept of an infinite derivative – i.e. DF(x) = ±∞ – is seldom studied due to a plethora of complications that arise from this definition. For instance, in this extended sense, algebraic expressions involving derivatives are generally undefined; and two continuous functions possessing identical derivatives at every point of an interval generally differ by a non-constant function. These problems are fundamentally irremediable insofar as calculus is concerned and must therefore be addressed in a more general setting. This is quite difficult since the literature on infinite derivatives is rather sparse and seldom accessible to non-specialists. Therefore we supply a self-contained thesis on continuous functions with infinite derivatives aimed at graduate students with a background in real analysis and measure theory.  Predominately we study continuous primitives which satisfy the Luzin condition (N) by establishing a deep connection with the strong Luzin condition – a weak form of absolute continuity which has its origins in the Henstock–Kurzweil theory of integration. The main result states that a function satisfies the strong Luzin condition if and only if it can be expressed as a sum of two such primitives. Furthermore, we establish some pathological properties of continuous primitives which fail to satisfy the Luzin condition (N).

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In this manuscript, we develop a nite liquidity framework for two-asset markets. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity models. We adopt Euler- Maruyama and Milstein scheme in the simulation of asset prices. Exchange and Spread option values are numerically estimated by Monte Carlo with the Margrabe option as a controlled variate. The time complexity of these numerical schemes is included. Finally, we provide some deep learning frameworks to implement these pricing models effectively. / Thesis / Master of Science (MSc)

Interfaces are ubiquitous in our daily life. A good understanding of the interfacial properties between different materials, or a single material in different physical states is of critical importance for us to explore the current world and bring benefits to mankind. In this work, interfacial behavior was investigated with the help of surface analysis techniques, such as quartz crystal microbalance with dissipation monitoring (QCM-D), surface plasmon resonance (SPR) and atomic force microscopy (AFM), in order to gain better understanding on biofuel conversion, gene/drug delivery, and chemical fixation of CO2. Biomimetic chelator-mediated Fenton (CMF) non-enzymatic degradations on cellulose and chitin thin films was studied by liquid-phase QCM-D and AFM. QCM-D is a powerful tool to monitor the kinetics of hydrolysis of regenerated cellulose and chitin model surfaces. Results from QCM-D and AFM showed that the majority of the biomass of the two model surfaces can be hydrolyzed by the CMF system. The initial degradation rates for both model surfaces by the CMF system are faster than that of the corresponding enzyme systems. The CMF system, which is a good non-enzymatic pretreatment agent for cellulose and chitin, may work on a wide variety of polysaccharide systems. Adsorption of cationic cellulose derivatives onto self-assembled monolayer (SAM) surfaces was investigated using liquid-phase SPR. Results from SPR showed that depending upon the cellulose derivative structure, irreversible adsorption ranging from a monolayer to ~1.6 layers of cellulose derivative were formed on the SAM-COOH surface based upon a charge neutralization mechanism. At low salt concentrations, the long-range electrostatic attraction between the cationic cellulose derivatives (6-PyrCA and 6-MeIMCA) and the SAM surfaces facilitates the formation of a 2-dimensional monolayer. While, for TMACE, the energy gained through the hydrophobic interaction between adjacent long polyelectrolyte branches may afford the electrostatic repulsion and chain entropy penalties, resulting in the formation of 3-dimensional adsorbed polyelectrolyte layers. Adsorption of 1,2-epoxybutane gas molecules onto/into VPI-100 metal–organic frameworks (MOFs) was studied by gas-phase QCM-D experiments. Results from QCM-D demonstrated that VPI-100 (Ni) MOFs have higher irreversible adsorption per unit cell (θ) and faster diffusion coefficients (D) than VPI-100 (Cu) MOFs. The presence of bound counter-balancing ions on the metallo-cyclam core was attributed as the cause of the higher θ and faster D through the Ni analogue, which suggests the MOF-epoxide interaction occurs at the metallo-cyclam. This study shed light upon tuning MOF structures for better CO2 sorption and epoxide activation to gain higher catalytic efficiency. Finally, in operando high energy X-ray diffraction (HEXRD) was used to monitor the phase transition of the NaxNi1/3Co1/3Mn1/3O2 cathode material during the sintering process. The first charge/discharge cycle of the NaxNi1/3Co1/3Mn1/3O2 cathode materials in different phases were also studied by in operando HEXRD. It was found that the intergrowth P2/O1/O3 cathode (NCM-Q cathode) can inhibit the irreversible P2–O2 phase transition and simultaneously improve the structural stability of the O3 and O1 phases during cycling. The NCM-Q cathode with triple-phase integration demonstrates highly reversible phase evolution during high voltage cycling, possibly leading to a highly reversible capacity and good cycle stability. / Doctor of Philosophy / Interfaces and surfaces are everywhere. Many critical processes, such as molecular recognition, catalysis, and charge transfer, take place at interfaces. The surfaces of plants and animals provide barriers from pathogens, prevent damage from mechanical impacts, detect external stimuli, etc. Inside the human body, nutrition and oxygen are adsorbed through interactions between substances and cell surfaces. Investigations of interfacial behaviors may help us understand our current world better and bring benefits to mankind. In this dissertation, the interface between bio-renewable natural polymers and biomimetic chelators, the interface between a self-assembled monolayer and cationic cellulose derivatives, and the interface between metal–organic frameworks (MOF) and 1,2-epoxybutane gas molecules, were studied with a quartz crystal microbalance with dissipation monitoring (QCM-D), surface plasmon resonance (SPR) and atomic force microscopy (AFM), to gain insights into biofuel conversion, gene/drug delivery and chemical fixation of CO2, respectively. Additionally, thermally and electrochemically induced phase transitions in sodium-ion battery (SIB) cathode materials were probed via in operando high energy X-ray diffraction (HEXRD). Biomimetic chelator-mediated Fenton (CMF) non-enzymatic degradations of cellulose and chitin thin films were studied by liquid-phase QCM-D and AFM. It was found that the majority of the biomass of the two model surfaces can be degraded by the CMF system. Adsorption of cationic cellulose derivatives onto self-assembled monolayer (SAM) surfaces was investigated using liquid-phase SPR. It was found that both the absorbed layer conformation and the absorbed amount depend upon the interplay between long-range electrostatic interactions and short-range interactions. Adsorption of 1,2-epoxybutane gas molecules onto/into VPI-100 MOFs was studied by gas-phase QCM-D experiments. Data from QCM-D revealed the irreversible gas molecule absorption onto/into MOFs and shed light upon tuning MOF structures for better CO2 sorption and epoxide activation to gain higher catalytic efficiency. Finally, the in operando high energy X-ray diffraction (HEXRD) was used to probe thermally and electrochemically induced phase transitions in sodium-ion battery (SIB) cathode materials. It was found that the NCM-Q cathode with triple-phase integration demonstrates highly reversible phase evolution during high voltage cycling, possibly leading to a highly reversible capacity and good cycle stability.