Optimal stopping and mathematical finance are intimately connected since the value of an American option is given as the solution to an optimal stopping problem. Such a problem can be viewed as a game in which we are trying to maximize an expected reward. The solution involves finding the best possible strategy, or equivalently, an optimal stopping time for the game. Moreover, the reward corresponding to this optimal time should be determined. It is also of interest to know how the solution depends on the model parameters. For example, when pricing and hedging an American option, the volatility needs to be estimated and it is of great practical importance to know how the price and hedging portfolio are affected by a possible misspecification. The first paper of this thesis investigates the performance of the delta hedging strategy for a class of American options with non-convex payoffs. It turns out that an option writer who overestimates the volatility will obtain a superhedge for the option when using the misspecified hedging portfolio. In the second paper we consider the valuation of a so-called stock loan when the lender is allowed to issue a margin call. We show that the price of such an instrument is equivalent to that of an American down-and-out barrier option with a rebate. The value of this option is determined explicitly together with the optimal repayment strategy of the stock loan. The third paper considers the problem of how to optimally stop a Brownian bridge. A finite horizon optimal stopping problem like this can rarely be solved explicitly. However, one expects the value function and the optimal stopping boundary to satisfy a time-dependent free boundary problem. By assuming a special form of the boundary, we are able to transform this problem into one which does not depend on time and solving this we obtain candidates for the value function and the boundary. Using stochastic calculus we then verify that these indeed satisfy our original problem. In the fourth paper we consider an investor wanting to take advantage of a mispricing in the market by purchasing a bull spread, which is liquidated in case of a market downturn. We show that this can be formulated as an optimal stopping problem which we then, using similar techniques as in the third paper, solve explicitly. In the fifth and final paper we study convexity preservation of option prices in a model with jumps. This is done by finding a sufficient condition for the no-crossing property to hold in a jump-diffusion setting.

This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven. / QC20100621

We propose a dynamic programming (DP) approach for pricing American options over a finite time horizon. We model uncertainty in stock price that follows geometric Brownian motion (GBM) and let interest rate and volatility be fixed. A procedure based on dynamic programming combined with piecewise linear interpolation approximation is developed to price the value of options. And we introduce the free boundary problem into our model. Numerical experiments illustrate the relation between value of option and volatility.

<p>Consider a mixture of two immiscible, incompressible fluids e.g. oil and water. Since the fluids do not mix, an interface between the two fluids will form and move in time. The motion of the two fluids can be modelled by the incompressible Navier-Stokes equations for two phase flow with surface tension together with a representation of the moving interface. The parameters in the Navier-Stokes equations will depend on the position and other properties of the interface. The interface should move with the velocity of the flow at the interface. Since the fluids are incompressible, the density of each fluid is constant. Mass conservation then implies that the volume occupied by each of the two fluids should not change with time. The object of this thesis has been to develop a new numerical method to simulate incompressible two phase flow accurately that conserves mass and volume of each fluid correctly.</p><p>Numerical simulations of incompressible two phase flow with surface tension have been a challenge for many years. Several methods have been developed and used prior to the work presented in this thesis. The two most commonly used methods are volume of fluid methods and level set methods. There are advantages and disadvantages of both of the methods.</p><p>In volume of fluid methods the interface is represented by a discontinuity of a globally defined function. Because of the discontinuity it is hard both to move the interface as well as to calculate properties of the interface such as curvature. Specially designed methods have to be used, and all these methods are low order accurate. Volume of fluid methods do however conserve the volumes of the two fluids correctly.</p><p>In level set methods the interface is represented by the zero contour of the globally defined signed distance function. This function is smooth across the interface. Since the function is smooth, standard methods for partial differential equations can be used to advect the interface accurately. A reinitialization is however needed to make sure that the level set function remains a signed distance function. During this process the zero contour might move slightly. Because of this, the volume conservation of the method becomes poor.</p><p>In this thesis we present a new level set method. The method is designed such that the volume of each fluid is conserved, at least approximately. The interface is represented by the 0.5 contour of a regularized characteristic function. As for standard level set methods, the interface is moved first by an advective step, and then reinitialized. Unlike traditional level set methods, we can formulate the reinitialization as a conservation law. Conservative methods can then be used to move and to reinitialize the level set function numerically. Since the level set function is a regularized characteristic function, we can expect good conservation of the volume bounded by the interface.</p><p>The method is discretized using both finite differences and finite elements. Uniform and adaptive grids are used in both two and three space dimensions. Good convergence as well as volume conservation is observed. Theoretical studies are performed to investigate the conservation and the computational time needed for reinitialization.</p>

Optimal control (motion planning) of the free interface in classical two-phase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The first-order optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function. Suitable discretization schemes for the forward and adjoint systems are described. Numerical examples verify the correctness and flexibility of the proposed scheme.

Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.

Les problèmes d'optimisation de forme sont présents naturellement en physique, ingénierie, biologie, etc. Ils visent à répondre à différentes questions telles que:-A quoi une aile d'avion parfaite pourrait ressembler?-Comment faire pour réduire la résistance d'un objet en mouvement dans un gaz ou un fluide?-Comment construire une structure élastique de rigidité maximale?-Quel est le comportement d'un système de cellules en interaction?Pour des exemples précis et autres applications de l'optimisation de forme nous renvoyons à [20] et [69]. Ici, nous traitons les aspects mathématiques théoriques de l'optimisation de forme, concernant l'existence d'ensembles optimaux ainsi que leur régularité. Dans toutes les situations que l'on considère, la fonctionnelle dépend de la solution d'une certaine équation aux dérivées partielles posée sur la forme inconnue. Nous allons parfois se référer à cette fonction comme une fonction d'état.Les fonctions d'état les plus simples, mais qui apparaissent dans beaucoup de problèmes, sont données par les solutions des équations -Δw = 1 et -Δu = λu,qui sont liées à la torsion et aux modes d'oscillation d'un objet donné. Notre étude se concentrera principalement sur ces fonctionnelles de formes, impliquant la torsion et le spectre.[20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005. / The shape optimization problems naturally appear in engineering and biology. They aim to answer questions as:-What a perfect wing may look like?-How to minimize the resistance of a moving object in a gas or a fluid?-How to build a rod of maximal rigidity?-What is the behaviour of a system of cells?The shape optimization appears also in physics, mainly in electrodynamics and in the systems presenting both classical and quantum mechanics behaviour. For explicit examples and furtheraccount on the applications of the shape optimization we refer to the books [20] and [69]. Here we deal with the theoretical mathematical aspects of the shape optimization, concerning existence of optimal sets and their regularity. In all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation. We will sometimes refer to this function as a state function.The simplest state functions are provided by solutions of the equations−∆w = 1 and −∆u = λu,which usually represent the torsion rigidity and the oscillation modes of a given object. Thus our study will be concentrated mainly on the situations, in which these state functions appear,i.e. when the optimality is intended with respect to energy and spectral functionals. [20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005.

Cette thèse présente des travaux concernant des phénomènes d'invasion tumorale, aux échelles tissulaire et cellulaire. La première partie est consacrée à deux modèles mathématiques continus. Le premier est un modèle macroscopique de croissance d'un cancer du sein qui se focalise sur la description du passage du stade in situ au stade invasif. Basé sur des équations d'advection d'espèces cellulaires, il tient compte de la géométrie et de l'éventuelle dégradation des tissus, dans le cas où la tumeur produit des enzymes protéolytiques qui permettent l'invasion. Le second modèle concerne le phénomène d'invadopodia, à l'échelle de la cellule. C'est un problème d'interface mobile qui décrit le changement de morphologie des cellules pré-métastatiques qui leur permet de dégrader les tissus pour migrer dans le reste de l'organisme. Chacun de ces deux modèles tient compte des couplages forts inhérents au phénomènes biologiques en jeu.La seconde partie est consacrée aux méthodes numériques développées pour résoudre ces deux problèmes et surmonter les difficultés liées aux couplages et non linéarités. Elles sont construites sur grille cartésienne uniforme, à partir des différences finies et d'une version stabilisée de la méthode Ghost fluid. Leur particularité est de tirer pleinement parti des propriétés de superconvergence de la solution du problème de Poisson, spécifiquement étudiées afin d'aboutir à la résolution des problèmes de cancer du sein et d'invadopodia à l'ordre un ou deux, en fonction de la précision désirée. Cesméthodes peuvent également être utilisées pour résoudre d'autres problèmes d'interface mobile. / In this thesis, we present a study about phenomena of tumor invasion, at the tissues and cell scales.The first part is devoted to two continuous mathematical models. The first one is a macroscopic model for breast cancer growth, which focuses on the transition between the stage in situ and the invasive phase of growth. This model is based on advection equations for cellular species. The geometry and possible tissue damage are taken into account. Invasion occurs when the tumor cells produce proteolytic enzymes. The second model deals with the phenomenon of invadopodia, at the cell scale.This is a free boundary problem, which describes the change in morphology of pre-metastatic cells,enabling them to degrade the tissues and migrate into the rest of the body. Each of these models reflects the strong coupling of biological phenomena.The second part is devoted to numerical methods specifically developed to solve these problems and overcome coupling and nonlinearities. They are built on uniform Cartesian grids, thanks to the finite difference method, and a stabilized version of the Ghost fluid method. Their peculiarity is to take full advantage of superconvergence properties of the Poisson problem solution. These properties are specifically studied, leading to the first or second order numerical computation of the problems ofbreast cancer and invadopodia, depending on the desired accuracy. These methods can also be used to solve other free boundary problems.

We study a free boundary problem with initial data given on three-dimensional cones. In particular, we study when the free boundary is allowed to pass through the vertex of the cone, which depends on the cone’s slope, determined by a parameter c. With a mix of analytical and computational methods, we show that the free boundary may pass through the vertex of the cone when c ≤ 0.43.

Consider a mixture of two immiscible, incompressible fluids e.g. oil and water. Since the fluids do not mix, an interface between the two fluids will form and move in time. The motion of the two fluids can be modelled by the incompressible Navier-Stokes equations for two phase flow with surface tension together with a representation of the moving interface. The parameters in the Navier-Stokes equations will depend on the position and other properties of the interface. The interface should move with the velocity of the flow at the interface. Since the fluids are incompressible, the density of each fluid is constant. Mass conservation then implies that the volume occupied by each of the two fluids should not change with time. The object of this thesis has been to develop a new numerical method to simulate incompressible two phase flow accurately that conserves mass and volume of each fluid correctly. Numerical simulations of incompressible two phase flow with surface tension have been a challenge for many years. Several methods have been developed and used prior to the work presented in this thesis. The two most commonly used methods are volume of fluid methods and level set methods. There are advantages and disadvantages of both of the methods. In volume of fluid methods the interface is represented by a discontinuity of a globally defined function. Because of the discontinuity it is hard both to move the interface as well as to calculate properties of the interface such as curvature. Specially designed methods have to be used, and all these methods are low order accurate. Volume of fluid methods do however conserve the volumes of the two fluids correctly. In level set methods the interface is represented by the zero contour of the globally defined signed distance function. This function is smooth across the interface. Since the function is smooth, standard methods for partial differential equations can be used to advect the interface accurately. A reinitialization is however needed to make sure that the level set function remains a signed distance function. During this process the zero contour might move slightly. Because of this, the volume conservation of the method becomes poor. In this thesis we present a new level set method. The method is designed such that the volume of each fluid is conserved, at least approximately. The interface is represented by the 0.5 contour of a regularized characteristic function. As for standard level set methods, the interface is moved first by an advective step, and then reinitialized. Unlike traditional level set methods, we can formulate the reinitialization as a conservation law. Conservative methods can then be used to move and to reinitialize the level set function numerically. Since the level set function is a regularized characteristic function, we can expect good conservation of the volume bounded by the interface. The method is discretized using both finite differences and finite elements. Uniform and adaptive grids are used in both two and three space dimensions. Good convergence as well as volume conservation is observed. Theoretical studies are performed to investigate the conservation and the computational time needed for reinitialization. / QC 20101122