The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

Le mouvement Brownien réfléchi de manière oblique dans le quadrant, introduit par Harrison, Reiman, Varadhan et Williams dans les années 80, est un objet largement analysé dans la littérature probabiliste. Cette thèse, qui présente l’étude complète de la mesure invariante de ce processus dans tous les cônes du plan, a pour objectif plus global d’étendre au cadre continu une méthode analytique développée initialement pour les marches aléatoires dans le quart de plan par Fayolle, Iasnogorodski et Malyshev dans les années 70. Cette approche est basée sur des équations fonctionnelles, reliant des fonctions génératrices dans le cas discret et des transformées de Laplace dans le cas continu. Ces équations permettent de déterminer et de résoudre des problèmes frontière satisfaits par ces fonctions génératrices. Dans le cas récurrent, cela permet de calculer explicitement la mesure invariante du processus avec rebonds orthogonaux, dans le chapitre 2, et avec rebonds quelconques, dans le chapitre 3. Les transformées de Laplace des mesures invariantes sont prolongées analytiquement sur une surface de Riemann induite par le noyau de l’équation fonctionnelle. L’étude des singularités et l’application de méthodes du point col sur cette surface permettent de déterminer l’asymptotique complète de la mesure invariante selon toutes les directions dans le chapitre 4. / Obliquely reflected Brownian motion in the quadrant, introduced by Harrison, Reiman, Varadhan and Williams in the eighties, has been studied a lot in the probabilistic literature. This thesis, which presents the complete study of the invariant measure of this process in all the cones of the plan, has for overall aim to extend to the continuous framework an analytic method initially developped for random walks in the quarter plane by Fayolle, Iasnogorodski and Malyshev in the seventies. This approach is based on functional equations which link generating functions in the discrete case and Laplace transform in the continuous case. These equations allow to determine and to solve boundary value problems satisfied by these generating functions. In the recurrent case, it permits to compute explicitly the invariant measure of the process with orthogonal reflexions, in the chapter 2, and with any reflexions, in the chapter 3. The Laplace transform of the invariant measure is analytically extended to a Riemann surface induced by the kernel of the functional equation. The study of singularities and the use of saddle point methods on this surface allows to determine the full asymptotics of the invariant measure along every directions in the chapter 4.

Dans cette thèse, on s'intéresse à deux problèmes provenant de l'étude mathématique des fluides incompressibles visqueux : la propagation de la régularité tangentielle et le mouvement d'une surface libre.La première question concerne plus particulièrement l'étude qualitative de l'évolution de quantités thermodynamiques telles que la température dans l'équation de Boussinesq sans diffusion et la densité dans le système de Navier-Stokes non homogène. Typiquement, on suppose que ces deux quantités sont, à l'instant initial, discontinues le long d'une interface à régularité h"oldérienne. Comme conséquence de résultats de propagation de régularité tangentielle pour le champ de vitesses, on établit que la régularité des interfaces persiste pour tout temps aussi bien en dimension deux d'espace, qu'en dimension supérieure (avec condition de petitesse). Notre approche suit celle du travail de J.-Y. Chemin dans les années 90 pour le problème des poches de tourbillon dans les fluides incompressiblesparfaits.Dans le cas présent, outre cette hypothèse de régularité tangentielle, nous n'avons besoin que d'une régularité critique sur le champ de vitesses.La démonstration repose sur le calcul para-différentiel et les espaces de multiplicateurs.Dans la dernière partie de la thèse, on considère le problème à frontière libre pour le système de Navier-Stokes incompressible à deux phases. Ce système permet de décrire l'évolution d'un mélange de deux fluides non miscibles tels que l'huile et l'eau par exemple. Différents cas de figure sont étudiés : le cas d'un réservoir borné, d'une goutte ou d'une rivière à profondeur finie.On établit l'existence et l'unicité à temps petit pour ce problème. Notre démonstration repose fortement sur des propriétés de régularité maximale parabolique de type $L_p$-$L_q / This thesis is dedicated to two different problems in the mathematical study of the viscous incompressible fluids: the persistence of tangential regularity and the motion of a free surface.The first problem concerns the study of the qualitative properties of some thermodynamical quantities in incompressible fluid models, such as the temperature for Boussinesq system with no diffusion and the density for the non-homogeneous Navier-Stokes system. Typically, we assume those two quantities to be initially piecewise constant along an interface with H"older regularity.As a consequence of stability of certain directional smoothness of the velocity field, we establish that the regularity of the interfaces persist globally with respect to time both in the two dimensional and higher dimensional cases (under some smallness condition). Our strategy is borrowed from the pioneering works by J.-Y.Chemin in 1990s on the vortex patch problem for ideal fluids.Let us emphasize that, apart from the directional regularity, we only impose rough (critical) regularity on the velocity field. The proof requires tools from para-differential calculus and multiplier space theory.In the last part of this thesis, we are concerned with the free boundary value problem for two-phase density-dependent Navier-Stokes system.This model is used to describe the motion of two immiscible liquids, like the oil and the water. Such mixture may occur in different situations, such as in a fixed bounded container, in a moving bounded droplet or in a river with finite depth. We establish the short time well-posedness for this problem. Our result strongly relies on the $L_p$-$L_q$ maximal regularity theoryfor parabolic equations

Microelectromechanical systems (MEMS) depend on mechanical deformation to sense their environment, enhance electrical circuitry, or store data. Nonlinear forces arising from multiphysics phenomena at the micro- and nanoscale -- van der Waals forces, electrostatic fields, dielectric charging, capillary forces, surface roughness, asperity interactions -- lead to challenging problems for analysis, simulation, and measurement of the deforming device elements. Herein, a foundation for the study of mechanical deformation is provided through computational and experimental studies of MEMS microcantilever capacitive switches. Numerical techniques are built to capture deformation equilibria expediently. A compact analytical model is developed from principle multiphysics governing operation. Experimental measurements support the phenomena predicted by the analytical model, and finite element method (FEM) simulations confirm device-specific performance. Altogether, the static multistability and quasistatic performance of the electrostatically-actuated switches are confirmed across analysis, simulation, and experimentation. <p><br></p> <p>The nonlinear multiphysics forces present in the devices are critical to the switching behavior exploited for novel applications, but are also a culprit in a common failure mode when the attractive forces overcome the restorative and repulsive forces to result in two elements sticking together. Quasistatic operation is functional for switching between multistable states during normal conditions, but is insufficient under such stiction-failure. Exploration of dynamic methods for stiction release is often the only option for many system configurations. But how and when is release achieved? To investigate the fundamental mechanism of dynamic release, an atomic force microscopy (AFM) system -- a microcantilever with a motion-controlled base and a single-asperity probe tip, measured and actuated via lasers -- is configured to replicate elements of a stiction-failed MEMS device. Through this surrogate, observable dynamic signatures of microcantilever deflection indicate the onset of detachment between the probe and a sample.</p>

[pt] Estudamos duas classes de equações diferenciais parciais, nomeadamente: uma equação de transferência radiativa e uma equação do calor duplamente não-linear. O primeiro modelo envolve uma equação não-local, na presença de um operador de espalhamento. Estuda-se a boa colocação do problema no semi-plano, no regime peaked. Prova-se um lema de averaging, que produz regularidade interior para o problema, além de regularização fracionária para as derivadas temporais da solução. O segundo conjunto de resultados da tese trata de uma equação de Trudinger com graus de não-linearidade distintos. Aproxima-se este problema pela p-equação do calor e importa-se regularidade da última para a primeira. Como consequência, mostra-se um resultado de regularidade melhorada no contexto não homogêneo. / [en] We consider two classes of partial differential equations. Namely: the radiative transfer equation and a doubly nonlinear model. The former concerns a nonlocal problema, driven by a scattering operator. We study the well-posedness of solutions in the peaked regime, for the half-space. A new averaging lemma yields interior regularity for the solutions and improved fractional regularization for the time derivatives. The second model we examine is a Trudinger equation with distinct nonlinearities degrees. Inspired by ideas launched by L. Caffarelli, we resort to approximation methods and prove improved regularity results for the solutions. The strategy is to relate our equation with p-caloric functions.

科学研究費補助金 研究種目:総合研究(A) 課題番号:04302008 研究代表者:三井 斌友 研究期間:1992-1994年度