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Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 145-151).

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.

It is the main goal of this thesis to study the regularity of solutions for a nonlinear elliptic system coming from population segregation, and the free boundary problem that is obtained in the limit as the competition parameter goes to infinity [mathematical symbol]. The main results are existence and Hölder regularity of solutions of the elliptic system, characterization of the limit as a free boundary problem, and Lipschitz regularity at the boundary for the limiting problem. / text

The focus of this thesis is optimal path planning. The path planning problem is posed as an optimal control problem, for which the viscosity solution to the static Hamilton-Jacobi-Bellman (HJB) equation is used to determine the optimal path. The Ordered Upwind Method (OUM) has been previously used to numerically approximate the viscosity solution of the static HJB equation for direction-dependent weights. The contributions of this thesis include an analytical bound on the convergence rate of the OUM for the boundary value problem to the viscosity solution of the HJB equation. The convergence result provided in this thesis is to our knowledge the tightest existing bound on the convergence order of OUM solutions to the viscosity solution of the static HJB equation. Only convergence without any guarantee of rate has been previously shown. Navigation functions are often used to provide controls to robots. These functions can suffer from local minima that are not also global minima, which correspond to the inability to find a path at those minima. Provided the weight function is positive, the viscosity solution to the static HJB equation cannot have local minima. Though this has been discussed in literature, a proof has not yet appeared. The solution of the HJB equation is shown in this work to have no local minima that is not also global. A path can be found using this method. Though finding the shortest path is often considered in optimal path planning, safe and energy efficient paths are required for rover path planning. Reducing instability risk based on tip-over axes and maximizing solar exposure are important to consider in achieving these goals. In addition to obstacle avoidance, soil risk and path length on terrain are considered. In particular, the tip-over instability risk is a direction-dependent criteria, for which accurate approximate solutions to the static HJB equation cannot be found using the simpler Fast Marching Method. An extension of the OUM to include a bi-directional search for the source-point path planning problem is also presented. The solution is found on a smaller region of the environment, containing the optimal path. Savings in computational time are observed. A comparison is made in the path planning problem in both timing and performance between a genetic algorithm rover path planner and OUM. A comparison in timing and number of updates required is made between OUM and several other algorithms that approximate the same static HJB equation. Finally, the OUM algorithm solving the boundary value problem is shown to converge numerically with the rate of the proven theoretical bound.

Defect generation and propagation in thin films, such as separation membranes, can lead to premature or catastrophic failure of devices such as polymer electrolyte membrane fuel cells (PEMFC). It is hypothesized that defects (e.g., air bubbles, pin-holes, and holes) originate during the manufacturing stage, if precise control is not maintained over the coating process, and they propagate during system operation. Experimental and numerical studies were performed to detect and eliminate defects that were induced during slot die coating of high-viscosity (1 to 40 Pa-s), shear-thinning solutions. The effects of fluid properties, geometric parameters and processing conditions on air entrainment and coating windows (limited set of processing conditions for which defect-free coating exists) were studied. When smaller slot gaps and coating gaps were used, relatively small bubbles were entrained in the coated film. The air bubble sizes increased as the viscosity of the coating solution decreased. A semi-empirical model correlating the maximum coating speed to a solution's material properties, geometric parameters and processing conditions was developed. Such a predictive model will enable engineers to determine the maximum coating boundary for shear-thinning and Newtonian solutions within certain constraints. Smaller coating gaps and low-viscosity solutions produced higher coating speeds. The surface tension property of the coating solution provided stability to the coating bead. Therefore, solutions with higher surface tension could be processed at higher coating speeds.

Nous nous intéressons à des résultats d'approximation et de régularité pour des solutions de viscosité d'équations elliptiques et paraboliques non-linéaires. Dans le chapitre 1, nous proposons, pour une classe générale d'équations elliptiques et paraboliques non-linéaires munies de conditions de Neumann inhomogènes, une interprétation de contrôle déterministe par des jeux répétés à deux personnes qui consiste à représenter la solution comme la limite de la suite des scores associés aux jeux. La condition de Neumann intervient par une pénalisation adaptée près de la frontière. En s'inspirant d'une approche abstraite proposée par Barles et Souganidis, nous prouvons la convergence en établissant des propriétés de monotonie, stabilité et consistance. Le chapitre 2 est consacré à des résultats de régularité sur les solutions d'équations paraboliques non-linéaires associés à un opérateur uniformément elliptique. Nous donnons une estimation de la mesure de Lebesgue de l'ensemble des points possédant un développement de Taylor quadratique global avec un contrôle sur la taille du terme cubique. Sous une hypothèse supplémentaire sur la régularité de la non-linéarité, nous en déduisons un résultat de régularité partielle höldérienne des solutions. Dans les chapitres 3 et 4, nous proposons une méthode générale pour obtenir des taux algébriques de convergence de solutions de schémas d'approximation vers la solution de viscosité sous l'hypothèse d'uniforme ellipticité de l'opérateur. Nous donnons un taux de convergence pour des schémas elliptiques obtenus par principe de programmation dynamique et nous prouvons un taux pour des schémas paraboliques par différences finies et implicites en temps. / In this thesis we study some approximation and regularity results for viscosity solutions of fully nonlinear elliptic and parabolic equations. In the first chapter, we consider a broad class of fully nonlinear elliptic and parabolic equations with inhomogeneous Neumann boundary conditions. We provide a deterministic control interpretation through two-person repeated games which represents the solution as the limit of the sequence of the scores associated to the games. The Neumann condition is modeled by a suitable penalization near the boundary. Inspiring by an abstract method of Barles and Souganidis, we prove the convergence of the score to the solution of the equation by establishing monotonicity, stability and consistency. The second chapter presents some regularity results about viscosity solutions of parabolic equations associated to a uniformly elliptic operator. First we obtain a Lebesgue measure estimate on the points having a quadratic Taylor expansion with a controlled cubic term. Under an additional assumption on the regularity of the nonlinearity, we deduce a partial regularity result about the Hölder regularity of these solutions. In the third and fourth chapters, we propose a general approach to determine algebraic rates of convergence of solutions of approximation schemes to the viscosity solution of fully nonlinear elliptic or parabolic equations under the assumption of uniform ellipticity of the operator. We first give the rate associated to the elliptic schemes derived by dynamic programming principles and proposed by Kohn and Serfaty. We then prove a rate of convergence for finite-difference schemes implicit in time associated to fully nonlinear parabolic equations.

Cette thèse porte sur la modélisation, l’analyse et l’analyse numérique des équations aux dérivées partielles non-linéaires et non-locales avec des applications au trafic routier. Le trafic routier peut être modélisé à des différentes échelles. En particulier, on peut considérer l’échelle microscopique qui décrit la dynamique de chaque véhicule individuellement et l’échelle macroscopique qui voit le trafic comme un fluide et qui décrit le trafic en utilisant des quantités macroscopiques comme la densité des véhicules et la vitesse moyenne. Dans cette thèse, en utilisant la théorie des solutions de viscosité, on fait le passage entre les modèles microscopiques et les modèles macroscopiques. L’intérêt de ce passage est que les modèles microscopiques sont plus intuitifs et faciles à manipuler pour simuler des situations particulières (bifurcations, feux tricolores,...) mais ils ne sont pas adaptés à des grosses simulations (pour simuler le trafic dans toute une ville par exemple). Au contraire, les modèles macroscopiques sont moins évidents à modifier (pour simuler une situation particulière) mais ils peuvent être utilisés pour des simulations à grande échelle. L’idée est donc de trouver le modèle macroscopique équivalent à un modèle microscopique qui décrit un scénario précis (une jonction, une bifurcation, des différents types de conducteurs, une zone scolaire,...). La première partie de cette thèse contient un résultat d’homogénéisation et d’homogénéisation numérique pour un modèle microscopique avec différents types de conducteurs. Dans une seconde partie, on obtient des résultats d’homogénéisation et d’homogénéisation numérique pour des modèles microscopiques con- tenant une perturbation locale (ralentisseur, zone scolaire,...). Finalement, on présente un résultat d’homogénéisation dans le cadre d’une bifurcation. / This work deals with the modelling, analysis and numerical analysis of non- linear and non-local partial differential equations and their application to traffic flow. Traffic can be simulated at different scales. Mainly, we have the microscopic scale which describes the dynamics of each of the vehicles individually and the macroscopic scale which describes the traffic as a fluid using macroscopic quantities such as the density of vehicles and the average speed. In this PhD thesis, using the theory of viscosity solutions, we derive macroscopic models from microscopic models. The interest of these results is that microscopic models are very intuitive and easy to manipulate to describe a particular situation (bifurcation, a traffic light,...), however, they are not adapted for big simulations (to simulate the traffic in an entire city for example). Conversely, macroscopic models are less easy to modify (to simulate a particular situation) but they can be used for big simulations. The idea is then to find the macroscopic model equivalent to a microscopic model describing a particular scenario (a junction, a bifurcation, different types of drivers, a school zone,...). The first part of this work contains an homogenization result and a numerical homogenization result for a microscopic model with different types of drivers. The second part contains an homogenization and numerical homogenization result for microscopic models with a local perturbation (a moderator, a school zone,...). Finally, we present an homogenization result for a bifurcation.

Le but de cette thèse est d'apporter une contribution à la problématique de valorisation de produits dérivés en marchés incomplets. Nous considérons tout d'abord les cibles stochastiques introduites par Soner et Touzi (2002) afin de traiter le problème de sur-réplication, et récemment étendues afin de traiter des approches plus générales par Bouchard, Elie et Touzi (2009). Nous généralisons le travail de Bouchard {\sl et al} à un cadre plus général où les diffusions sont sujettes à des sauts. Nous devons considérer dans ce cas des contrôles qui prennent la forme de fonctions non bornées, ce qui impacte de façon non triviale la dérivation des EDP correspondantes. Notre deuxième contribution consiste à établir une version des cibles stochastiques qui soit robuste à l'incertitude de modèle. Dans un cadre abstrait, nous établissons une version faible du principe de programmation dynamique géométrique de Soner et Touzi (2002), et nous dérivons, dans un cas d'EDS controllées, l'équation aux dérivées partielles correspondantes, au sens des viscosités. Nous nous intéressons ensuite à un exemple de couverture partielle sous incertitude de Knightian. Finalement, nous nous concentrons sur le problème de valorisation de produits dérivées {\sl hybrides} (produits dérivés combinant finance de marché et assurance). Nous cherchons plus particulièrement à établir une condition suffisante sous laquelle une règle de valorisation (populaire dans l'industrie), consistant à combiner l'approches actuarielle de mutualisation avec une approche d'arbitrage, soit valable.

The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated. In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows. Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows.