Dissertationen zum Thema „Free boundary model“

Abstract: A rigorous approach to Statistical Physics issues often produces interesting mathematical questions. This Ph.D. thesis is composed of two different parts. One does not intersect the other, but both research topics lie at the interface between Probability Theory and Statistical Mechanics. • In the first part we deal with reconstruction of a tree-indexedMarkov chain on Galton-Watson trees, improving previous bound byMossel and Peres, both for symmetric and strongly asymmetric chains. Moreover, we give some numerical estimates to compare our bound with those of other authors. We provide a sufficient condition of the form Q(d)c(M) < 1 for the non-reconstructability of tree-indexed q-stateMarkov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is a constant depending on the transition matrixM and Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. When considering the Potts model case we take this point of view too. Our theorem holds for possibly non-reversible M. In the case of the symmetric Ising model the method produces the correct reconstruction threshold, in the case of the (strongly) asymmetric Ising modelwhere the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds. • In the second part of the thesis we give sharp estimates for time uniformpropagation of chaos in some specialsmean field spin-flipmodels exhibiting phase transition. The first model is the dynamical Curie-Weiss model, that can be considered as the most basic mean field model. The second example is a model proposed recently in the context of credit risk in Finance; it describes the time evolution of finantial indicators for a network of interacting firms. Although we have chosen to deal with two specific models, the method we use appear to be rather general, and should work for other classes of models. A substantial limitation of our results is that they are limited to the subcritical case or, in StatisticalMechanical terms, to the high temperature regime.<br>Sommario: Un approccio rigoroso a questioni di Fisica Statistica spesso produce interessanti problemi matematici. Questa tesi di dottorato è composta da due parti. La prima non interseca la seconda, ma entrambe stanno sul confine tra Teoria della Probabilità e Meccanica Statistica. • La prima parte tratta il problema della ricostruzione per catene di Markov su alberi di tipo Galton-Watson. Miglioriamoi risultati precedentemente ottenuti da Mossel e Peres, sia per catene simmetriche che fortemente asimmetriche. Dimostriamo una condizione sufficiente della forma Q(d)c(M) < 1 per la non ricostruzione di catene diMarkov a q-stati sull’albero. Qui c(M) è una costante che dipende dalla matrice di transizione M e Q(d) è la media del numero di figli per vertice nell’albero di Galton-Watson. Questo risultato è equivalente alla purezza della misura libera di Gibbs. Quando consideriamo il caso del modello di Potts assumiamo anche questo punto di vista. Il teorema è valido anche per catene non reversibili. Nel caso del modello di Ising il nostro risultato produce la correta soglia di ricostruzione, nel caso di catene (fortemente) asimmetriche dove si sa che il bound di Kesten-Stigum non è esatto il metodo usato dà risultati numerici migliori. • Nella seconda parte diamo delle stime uniformi nel tempo per la propagazione del caos in alcuni modelli di spin con interazione a campo medio che presentano transizione di fase. Il primo è il modello dinamico di Curie-Weiss, che può essere considerato come il più semplice esempio di sistema con interazione a campo medio. Il secondo è un modello recentemente impiegato per spiegare i meccanismi del rischio di credito; esso descrive l’evoluzione temporale di indicatori finaziari per un gruppo di aziende interagenti quotate sul mercato. Anche se abbiamo trattato modelli specifici, crediamo che il metodo funzioni piuttosto in generale e che sia applicabile anche ad altre classi di modelli. Una limitazione sostanziale dei nostri risultati è che valgono solo nel caso sottocritico, che corrisponde, nel linguaggio della Meccanica Statistica, al regime di alta temperatura.

The main contribution of this thesis is to derive the properties and present a closed from solution of the exotic options under some specific types of Levy processes, such as American put options, American call options, British put options, British call options and American knock-out put options under either double exponential jump-diffusion processes or one-sided exponential jump-diffusion processes. Compared to the geometric Brownian motion, exponential jump-diffusion processes can better incorporate the asymmetric leptokurtic features and the volatility smile observed from the market. Pricing the option with early exercise feature is the optimal stopping problem to determine the optimal stopping time to maximize the expected options payoff. Due to the Markovian structure of the underlying process, the optimal stopping problem is related to the free-boundary problem consisting of an integral differential equation and suitable boundary conditions. By the local time-space formula for semi-martingales, the closed form solution for the options value can be derived from the free-boundary problem and we characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation arising from the early exercise premium (EEP) representation. Chapter 2 and Chapter 3 discuss American put options and American call options respectively. When pricing options with early exercise feature under the double exponential jump-diffusion processes, a non-local integral term will be found in the infinitesimal generator of the underlying process. By the local time-space formula for semi-martingales, we show that the value function and the optimal stopping boundary are the unique solution pair to the system of two integral equations. The significant contributions of these two chapters are to prove the uniqueness of the value function and the optimal stopping boundary under less restrictive assumptions compared to previous literatures. In the degenerate case with only one-sided jumps, we find that the results are in line with the geometric Brownian motion models, which extends the analytical tractability of the Black-Scholes analysis to alternative models with jumps. In Chapter 4 and Chapter 5, we examine the British payoff mechanism under one-sided exponential jump-diffusion processes, which is the first analysis of British options for process with jumps. We show that the optimal stopping boundaries of British put options with only negative jumps or British call options with only positive jumps can also be characterized as the unique solution to a nonlinear integral equation arising from the early exercise premium representation. Chapter 6 provides the study of American knock-out put options under negative exponential jump-diffusion processes. The conditional memoryless property of the exponential distribution enables us to obtain an analytical form of the arbitrage-free price for American knock-out put options, which is usually more difficult for many other jump-diffusion models.

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.

Mestrado em Mathematical Finance<br>Os problemas de apreçamento de opções têm sido um dos principais assuntos de em Matemática Financeira, desde a criação desse conceito nos anos 70. Mais especificamente, as opções americanas são de grande interesse nesta área do conhecimento porque são matematicamente muito mais complexas do que as opções europeias padrão e o modelo de Black-Scholes não fornece, na maioria dos casos, uma fórmula explícita para a determinação do preço deste tipo de opções. Nesta dissertação, mostramos como o estudo de opções americanas conduz à análise de problemas de fronteira livre devido à possibilidade de exercício antecipado, onde nosso principal objetivo é encontrar o preço de exercício ótimo. Também apresentamos a reformulação do problema em termos de um problema de complementaridade linear e de desigualdade variacional parabólica. Além disso, também abordamos a caracterização probabilística das opções americanas com base no conceito de tempos de paragem ótima. Essas formulações, aqui tratadas em termos analíticos ou probabilísticos, podem ser muito úteis na aplicação de métodos numéricos ao problema de precificação de opções do estilo americano, uma vez que, na maioria dos casos, é quase impossível encontrar soluções explícitas. Além disso, utilizamos o Método da Árvore Binomial, que é um método numérico muito simples do ponto de vista matemático, para ilustrar alguns aspectos da teoria estudada ao longo desta tese e para comparar as opções americanas com as opções europeias e bermudas, por meio de alguns exemplos numéricos.<br>Option pricing problems have been one of the main focuses in the field of Mathematical Finance since the creation of this concept in the 1970s. More specifically, American options are of great interest in this area of knowledge because they are much more complex mathematically than the standard European options and the Black-Scholes model cannot give an explicit formula to value this style options in most cases. In this dissertation, we show how pricing American options leads to free boundary problems because of the possibility of early exercise, where our main goal is to find the optimal exercise price. We also present how to reformulate the problem into a linear complementarity problem and a parabolic variational inequality. Moreover, we also address the probabilistic characterization of American options based on the concept of stopping times. These formulations, here viewed from the analytical and probabilistic point of view, can be very useful for applying numerical methods to the problem of pricing American style options since, in most cases, it is almost impossible to find explicit solutions. Furthermore, we use the Binomial Tree Method, which is a very simple numerical method from the mathematical point of view, to illustrate some aspects of the theory studied throughout this thesis and to compare American options with European and Bermudan Options, by means of a few numerical examples.<br>info:eu-repo/semantics/publishedVersion

Abstract Electrostatic interactions are very important in biomolecular systems. Electrostatic forces have received a great deal of attention due to their long-range nature and the trade-off between desolvation and interaction effects. It remains a challenging task to study and to predict the effects of electrostatic interactions in biomolecular systems. Computer simulation techniques that account for such interactions are an important tool for the study of biomolecular electrostatics. This study is largely concerned with the role of electrostatic interactions in biomolecular systems and with developing novel models to estimate the strength of such interactions. First, a novel formulation based upon continuum electrostatics to compute the electrostatic potential in and around two biomolecules in a solvent with ionic strength is presented. Many, if not all, current methods rely on the (non)linear Poisson-Boltzmann equation to include ionic strength. The present formulation, however, describes ionic strength through the inclusion of explicit ions, which considerably extends its applicability and validity range. The method relies on the boundary element method (BEM) and results in two very similar coupled integral equations valid on the dielectric boundaries of two molecules, respectively. This method can be employed to estimate the total electrostatic energy of two protein molecules at a given distance and orientation in an electrolyte solution with zero to moderately high ionic strength. Secondly, to be able to study interactions between biomolecules and membranes, an alternative model partly based upon the analytical continuum electrostatics (ACE) method has been also formulated. It is desirable to develop a method for calculating the total solvation free energy that includes both electrostatic and non-polar energies. The difference between this model and other continuum methods is that instead of determining the electrostatic potential, the total electrostatic energy of the system is calculated by integrating the energy density of the electrostatic field. This novel approach is employed for the calculation of the total solvation free energy of a system consisting of two solutes, one of which could be an infinite slab representing a membrane surface.

This thesis presents a numerical exploration of the dynamics governing rotating flow driven by a surface stress in the " sliced cylinder " model of Pedlosky & Greenspan (1967) and Beardsley (1969), and its close relative, the " sliced cone " model introduced by Griffiths & Veronis (1997). The sliced cylinder model simulates the barotropic wind-driven circulation in a circular basin with vertical sidewalls, using a depth gradient to mimic the effects of a gradient in Coriolis parameter. In the sliced cone the vertical sidewalls are replaced by an azimuthally uniform slope around the perimeter of the basin to simulate a continental slope. Since these models can be implemented in the laboratory, their dynamics can be explored by a complementary interplay of analysis and numerical and laboratory experiments. ¶ In this thesis a derivation is presented of a generalised quasigeostrophic formulation which is valid for linear and moderately nonlinear barotropic flows over large-amplitude topography on an f-plane, yet retains the simplicity and conservation properties of the standard quasigeostrophic vorticity equation (which is valid only for small depth variations). This formulation is implemented in a numerical model based on a code developed by Page (1982) and Becker & Page (1990). ¶ The accuracy of the formulation and its implementation are confirmed by detailed comparisons with the laboratory sliced cylinder and sliced cone results of Griffiths (Griffiths & Kiss, 1999) and Griffiths & Veronis (1997), respectively. The numerical model is then used to provide insight into the dynamics responsible for the observed laboratory flows. In the linear limit the numerical model reveals shortcomings in the sliced cone analysis by Griffiths & Veronis (1998) in the region where the slope and interior join, and shows that the potential vorticity is dissipated in an extended region at the bottom of the slope rather than a localised region at the east as suggested by Griffiths & Veronis (1997, 1998). Welander's thermal analogy (Welander, 1968) is used to explain the linear circulation pattern, and demonstrates that the broadly distributed potential vorticity dissipation is due to the closure of geostrophic contours in this geometry. ¶ The numerical results also provide insight into features of the flow at finite Rossby number. It is demonstrated that separation of the western boundary current in the sliced cylinder is closely associated with a " crisis " due to excessive potential vorticity dissipation in the viscous sublayer, rather than insufficient dissipation in the outer western boundary current as suggested by Holland & Lin (1975) and Pedlosky (1987). The stability boundaries in both models are refined using the numerical results, clarifying in particular the way in which the western boundary current instability in the sliced cone disappears at large Rossby and/or Ekman number. A flow regime is also revealed in the sliced cylinder in which the boundary current separates without reversed flow, consistent with the potential vorticity " crisis " mechanism. In addition the location of the stability boundary is determined as a function of the aspect ratio of the sliced cylinder, which demonstrates that the flow is stabilised in narrow basins such as those used by Beardsley (1969, 1972, 1973) and Becker & Page (1990) relative to the much wider basin used by Griffiths & Kiss (1999). ¶ Laboratory studies of the sliced cone by Griffiths & Veronis (1997) showed that the flow became unstable only under anticyclonic forcing. It is shown in this thesis that the contrast between flow under cyclonic and anticyclonic forcing is due to the combined effects of the relative vorticity and topography in determining the shape of the potential vorticity contours. The vorticity at the bottom of the sidewall smooths out the potential vorticity contours under cyclonic forcing, but distorts them into highly contorted shapes under anticyclonic forcing. In addition, the flow is dominated by inertial boundary layers under cyclonic forcing and by standing Rossby waves under anticyclonic forcing due to the differing flow direction relative to the direction of Rossby wave phase propagation. The changes to the potential vorticity structure under strong cyclonic forcing reduce the potential vorticity changes experienced by fluid columns, and the flow approaches a steady free inertial circulation. In contrast, the complexity of the flow structure under anticyclonic forcing results in strong potential vorticity changes and also leads to barotropic instability under strong forcing. ¶ The numerical results indicate that the instabilities in both models arise through supercritical Hopf bifurcations. The two types of instability observed by Griffiths & Veronis (1997) in the sliced cone are shown to be related to the western boundary current instability and " interior instability " identified by Meacham & Berloff (1997). The western boundary current instability is trapped at the western side of the interior because its northward phase speed exceeds that of the fastest interior Rossby wave with the same meridional wavenumber, as discussed by Ierley & Young (1991). ¶ Numerical experiments with different lateral boundary conditions are also undertaken. These show that the flow in the sliced cylinder is dramatically altered when the free-slip boundary condition is used instead of the no-slip condition, as expected from the work of Blandford (1971). There is no separated jet, because the flow cannot experience a potential vorticity " crisis " with this boundary condition, so the western boundary current overshoots and enters the interior from the east. In contrast, the flow in the sliced cone is identical whether no-slip, free-slip or super-slip boundary conditions are applied to the horizontal flow at the top of the sloping sidewall, except in the immediate vicinity of this region. This insensitivity results from the extremely strong topographic steering near the edge of the basin due to the vanishing depth, which demands a balance between wind forcing and Ekman pumping on the upper slope, regardless of the lateral boundary condition. The sensitivity to the lateral boundary condition is related to the importance of lateral friction in the global vorticity balance. The integrated vorticity must vanish under the no-slip condition, so in the sliced cylinder the overall vorticity budget is dominated by lateral viscosity and Ekman friction is negligible. Under the free-slip condition the Ekman friction assumes a dominant role in the dissipation, leading to a dramatic change in the flow structure. In contrast, the much larger depth variation in the sliced cone leads to a global vorticity balance in which Ekman friction is always dominant, regardless of the boundary condition.

Numerical simulations of incompressible, two-dimensional, monochromatically and bichromatically forced laminar free shear layers are performed on the basis of a vorticity-velocity formulation of the complete Navier-Stokes equations employing central finite differences. Spatially periodic shear layers developing in time (temporal model) are compared with shear layers developing in the stream-wise direction (spatial model). The regimes of linear growth and saturation of the fundamental are quantitatively scrutinized, the saturation of the subharmonic and vortex merging are investigated, and the effects of a forcing phase-shift between fundamental and subharmonic. For the spatial model the appearance of an unforced subharmonic was also examined. It was found that contrary to temporal shear layers a significant control of vortex merging by means of a forcing phase-shift and vortex shredding are not possible in spatial shear layers due to strong dispersion.

Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2010.<br>Committee Chair: Jeffrey A. Davis; Committee Co-Chair: James D. Meindl; Committee Member: Azad J. Naeemi; Committee Member: Dennis W. Hess; Committee Member: George F. Riley; Committee Member: Linda S. Milor. Part of the SMARTech Electronic Thesis and Dissertation Collection.

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.<br>MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

In this work we address to the problems of the loan origination decision-making systems. In accordance with the basic principles of the loan origination process we considered the main rules of a clients parameters estimation, a change-point problem for the given data and a disorder moment detection problem for the real-time observations. In the first part of the work the main principles of the parameters estimation are given. Also the change-point problem is considered for the given sample in the discrete and continuous time with using the Maximum likelihood method. In the second part of the work the disorder moment detection problem for the real-time observations is considered as a disorder problem for a non-homogeneous Poisson process. The corresponding optimal stopping problem is reduced to the free-boundary problem with a complete analytical solution for the case when the intensity of defaults increases. Thereafter a scheme of the real time detection of a disorder moment is given.

La motilité cellulaire est un phénomène impliqué dans de nombreux processus biologiques comme la propagation des cancers, la réponse immunitaire, la cicatrisation ou le développement embryonnaire. Ce phénomène est assuré par la capacité d'une cellule à se déformer d'une configuration symétrique, non polarisée, à une configuration asymétrique, polarisée, et à maintenir cette configuration asymétrique. Dans cette thèse, nous nous intéressons plus particulièrement au rôle du noyau dans ce phénomène. Pour cela, un modèle à frontière libre en dimension 2 est introduit. La cellule est modélisée par un fluide incompressible comportant une structure rigide modélisant le noyau. Des marqueurs de polarité sont présents dans le fluide et ceux-ci transduisent de manière active les forces appliquées par et sur le cytosquelette. Le modèle proposé permet également de modéliser l'effet de l'undercooling et de l'environnement extérieur sur la motilité cellulaire. Les différentes composantes du modèle sont étudiées de manière séparée dans la thèse.Nous étudions l'influence du noyau sur la motilité cellulaire. Pour cela nous considérons différents modèles dont un modèle déformable, un rigide et deux modèles jouets. Nous montrons que ces modèles admettent des états stationnaires. Via l'analyse linéaire de la stabilité, nous montrons qu'il existe un seuil à partir duquel l'état stationnaire radialement symétrique est instable. Pour chacun de ces modèles un schéma numérique aux éléments finis est développé. Les résultats numériques obtenus permettent de mettre en avant le lien entre la position du noyau dans la cellule et la polarisation de la cellule. Ils sont qualitativement en accord avec les observations biologiques. L'analyse des trajectoires est également réalisée. Un modèle analogue en dimension 1 qui est une équation non-locale et non-linéaire de Fokker-Planck est introduit. Nous montrons que celui-ci est bien posé. L'existence d'états stationnaires et d'ondes progressives est également étudiée. Un second modèle en dimension 1 basé sur une seconde modélisation est étudié.L'effet de l'undercooling sur la motilité cellulaire est étudié. En montrant l'existence d'ondes progressives et d'états stationnaires, nous illustrons que celui-ci a un effet stabilisant. L'existence d'ondes progressives est prouvée via un théorème de bifurcation. Nous montrons également à l'aide de l'analyse linéaire de la stabilité qu'il existe un seuil à partir duquel l'état stationnaire est instable.L'influence de la présence de signaux extérieurs attractifs sur la motilité cellulaire est aussi investiguée. Nous montrons l'existence d'un état stationnaire et l'existence d'une gamme de paramètres pour laquelle celui-ci est stable et une autre pour laquelle il est instable. Numériquement, nous illustrons qu'il existe une compétition entre forces induites par les marqueurs de polarité et celles induites par le signal extérieur.Cette thèse comporte également un travail effectué durant l'école d'été CEMRACS 2022. Un modèle d'un des mécanismes ayant lieu à la membrane, l'endocytose, y est présenté. Après avoir étudié le modèle, un schéma aux volumes finis est présenté. Celui-ci permet d'obtenir des résultats en accord avec les résultats biologiques<br>Cell motility is a phenomenon involved in many biological processes such as cancer propagation, immune response, wound healing and embryonic development. This phenomenon is ensured by a cell's ability to deform from a symmetrical, non-polarised configuration to an asymmetrical, polarised configuration, and to maintain this asymmetrical configuration. In this thesis, we focus on the role of the nucleus in this phenomenon. To this end, a free boundary model in dimension 2 is introduced. The cell is modelled by an incompressible fluid with a rigid structure modelling the nucleus. Polarity markers are present in the fluid and these actively transduce the forces applied by and on the cytoskeleton. The proposed model also models the effect of undercooling and the external environment on cell motility. The different components of the model are studied separately in the thesis.We are studying the influence of the nucleus on cell motility. We consider various models, including a deformable model, a rigid model and two toy models. We prove that these models admit stationary states. Using linear stability analysis, we demonstrate that there exists a threshold above which the radially symmetric stationary state is unstable. For each of these models, a finite element numerical scheme is developed. The numerical results obtained highlight the link between the position of the nucleus in the cell and the polarisation of the cell. They are in qualitative agreement with biological observations. Trajectory analysis is also carried out. An analogous model in dimension 1, which is a non-local and non-linear Fokker-Planck equation, is introduced. We show that it is well posed. The existence of stationary states and travelling waves is also studied. A second model in dimension 1 based on a second modelling is studied.The effect of undercooling on cell motility is studied. By proving the existence of travelling waves and stationary states, we illustrate that undercooling has a stabilising effect. The existence of travelling waves is proved using a bifurcation theorem. We also demonstrate, using linear stability analysis, that there exists a threshold above which the stationary state is unstable.The influence of the presence of attractive external signals on cell motility is also investigated. We prove the existence of a stationary state and the existence of a range of parameters for which it is stable and another for which it is unstable. Numerically, we illustrate that there is competition between the forces induced by the polarity markers and those induced by the external signal.This thesis also includes a work carried out during the CEMRACS 2022 summer school. A model of one of the mechanisms taking place at the membrane—endocytosis—is presented. After studying the model, a finite volume scheme is presented. This provides results that are consistent with biological results

Les marchés financiers ont connu, grâce aux études réalisées durant les trois dernières décennies, une expansion considérable et ont vu l'apparition de produits dérivés divers et variés. Parmi les plus répandus, on retrouve les options américaines. Une option américaine est par définition une option qu'on a le droit d'exercer avant l'échéance convenue T. Les plus basiques sont le Put ou le Call américain (respectivement option de vente (K - x)+ ou d'achat (x - K)+). La première partie, et la plus conséquente, de cette thèse est consacrée à l'étude des options américaines dans des modèles exponentiels de Lévy. On commence dans un cadre multidimensionnel caractérise le prix d'une option américaine, dont le Pay-off appartient à une classe de fonctions non forcément bornées, à l'aide d'une inéquation variationnelle au sens des distributions. On étudie, ensuite, les propriétés générales de la région d'exercice ainsi que de la frontière libre. On affine encore ces résultats en étudiant, en particulier, la région d'exercice d'un Call américain sur un panier d'actifs, où on caractérise en particulier la région d'exercice limite (à l'échéance). Dans un deuxième temps, on se place dans un cadre unidimensionnel et on étudie le comportement du prix critique (fonction délimitant la région d'exercice) d'un Put américain près de l'échéance. Particulièrement, on considère le cas où le prix ne converge pas vers le strike K, dans un modèle Jump-diffusion puis dans un modèle où le processus de Lévy est à saut pur avec un comportement proche de celui d'un &-stable. La deuxième partie porte sur l'approximation numérique de la Credit Valuation Adjustment (CVA). On y présente une méthode basée sur le calcul de Malliavin inspirées de celles utilisées pour les options américaines. Une étude de la complexité de cette méthode y est aussi présentée et comparée aux méthodes purement Monte Carlo et aux méthodes fondées sur la régression.

This thesis concerns the mathematical model of moving fluid interfaces in a Hele-Shaw cell: an experimental device in which fluid flow is studied by sandwiching the fluid between two closely separated plates. Analytic and numerical methods are developed to gain new insights into interfacial stability and bubble evolution, and the influence of different boundary effects is examined. In particular, the properties of the velocity-dependent kinetic undercooling boundary condition are analysed, with regard to the selection of only discrete possible shapes of travelling fingers of fluid, the formation of corners on the interface, and the interaction of kinetic undercooling with the better known effect of surface tension. Explicit solutions to the problem of an expanding or contracting ring of fluid are also developed.

Les écoulements en eau peu profonde se rencontrent dans de nombreuses situations d’intérêts : écoulements de rivières et dans les lacs, mais aussi dans les mers et océans (courants de marée, tsunami, etc.). Ils sont modélisés par un système d’équations aux dérivées partielles, où les inconnues sont la vitesse de l’écoulement et la hauteur d’eau. On peut supposer que la composante verticale de la vitesse est petite devant les composantes horizontales et que ces dernières sont indépendantes de la profondeur. Le modèle est alors donné par les équations de shallow water (SWEs). Cette thèse se concentre sur la conception d’une nouvelle technique d’interaction de plusieurs grilles imbriquées pour modèle en eau peu profonde en utilisant des méthodes numériques. La première partie de cette thèse comprend, La dérivation complète de ces équations à partir des équations de Navier- Stokes est expliquée. Etudier le développement et l’évaluation des méthodes numériques en utilisant des méthodes de différences finies et plusieurs exemples numériques sont appliqués utilisant la condition initiale du niveau gaussien pour 2DSWEs. Dans la deuxième partie de la thèse, nous sommes intéressés à proposer une nouvelle technique d’interaction de plusieurs grilles imbriquées pour résoudre les modèles océaniques en utilisant quatre choix des opérateurs de restriction avec des résultats de haute précision. Notre travail s’est concentré sur la résolution numérique de SWE par grilles imbriquées. A chaque niveau de résolution, nous avons utilisé une méthode classique de différences finies sur une grille C d’Arakawa, avec un schéma de leapfrog complété par un filtre d’Asselin. Afin de pouvoir affiner les calculs dans les régions perturbées et de les alléger dans les zones calmes, nous avons considéré plusieurs niveaux de résolution en utilisant des grilles imbriquées. Ceci permet d’augmenter considérablement le rapport performance de la méthode, à condition de régler efficacement les interactions (spatiales et temporelles) entre les grilles. Dans la troisième partie de cette thèse, plusieurs exemples numéériques sont testés pour 2DSWE avec imbriqués 3:1 et 5:1. Finalement, la quatrième partie de ce travail, certaines applications de grilles imbriquées pour le modèle tsunami sont présentées<br>Most flows in the rivers, seas, and ocean are shallow water flow in which the horizontal length andvelocity scales are much larger than the vertical ones. The mathematical formulation of these flows, so called shallow water equations (SWEs). These equations are a system of hyperbolic partial differentialequations and they are effective for many physical phenomena in the oceans, coastal regions, riversand canals. This thesis focuses on the design of a new two-way interaction technique for multiple nested grids 2DSWEs using the numerical methods. The first part of this thesis includes, proposing several ways to develop the derivation of shallow water model. The complete derivation of this system from Navier-Stokes equations is explained. Studying the development and evaluation of numerical methods by suggesting new spatial and temporal discretization techniques in a standard C-grid using an explicit finite difference method in space and leapfrog with Robert-Asselin filter in time which are effective for modeling in oceanic and atmospheric flows. Several numerical examples for this model using Gaussian level initial condition are implemented in order to validate the efficiency of the proposed method. In the second part of our work, we are interested to propose a new two-way interaction technique for multiple nested grids to solve ocean models using four choices of higher restriction operators (update schemes) for the free surface elevation and velocities with high accuracy results. Our work focused on the numerical resolution of SWEs by nested grids. At each level of resolution, we used explicit finite differences methods on Arakawa C-grid. In order to be able to refine the calculations in troubled regions and move them into quiet areas, we have considered several levels of resolution using nested grids. This makes it possible to considerably increase the performance ratio of the method, provided that the interactions (spatial and temporal) between the grids are effectively controlled. In the third part of this thesis, several numerical examples are tested to show and verify twoway interaction technique for multiple nested grids of shallow water models can works efficiently over different periods of time with nesting 3:1 and 5:1 at multiple levels. Some examples for multiple nested grids of the tsunami model with nesting 5:1 using moving boundary conditions are tested in the fourth part of this work

Ausgehend von einem thermodynamisch konsistenten Modell von Dreyer und Duderstadt für Tropfenbildung in Galliumarsenid-Kristallen, das Oberflächenspannung und Spannungen im Kristall berücksichtigt, stellen wir zwei mathematische Modelle zur Evolution der Größe flüssiger Tropfen in Kristallen auf. Das erste Modell behandelt das Regime diffusionskontrollierter Interface-Bewegung, während das zweite Modell das Regime Interface-kontrollierter Bewegung des Interface behandelt. Unsere Modellierung berücksichtigt die Erhaltung von Masse und Substanz. Diese Modelle verallgemeinern das wohlbekannte Mullins-Sekerka-Modell für die Ostwald-Reifung. Wir konzentrieren uns auf arsenreiche kugelförmige Tropfen in einem Galliumarsenid-Kristall. Tropfen können mit der Zeit schrumpfen bzw. wachsen, die Tropfenmittelpunkte sind jedoch fixiert. Die Flüssigkeit wird als homogen im Raum angenommen. Aufgrund verschiedener Skalen für typische Distanzen zwischen Tropfen und typischen Radien der flüssigen Tropfen können wir formal so genannte Mean-Field-Modelle herleiten. Für ein Modell im diffusionskontrollierten Regime beweisen wir den Grenzübergang mit Homogenisierungstechniken unter plausiblen Annahmen. Diese Mean-Field-Modelle verallgemeinern das Lifshitz-Slyozov-Wagner-Modell, welches rigoros aus dem Mullins-Sekerka-Modell hergeleitet werden kann, siehe Niethammer et al., und gut verstanden ist. Mean-Field-Modelle beschreiben die wichtigsten Eigenschaften unseres Systems und sind gut für Numerik und für weitere Analysis geeignet. Wir bestimmen mögliche Gleichgewichte und diskutieren deren Stabilität. Numerische Resultate legen nahe, wann welches der beiden Regimes gut zur experimentellen Situation passen könnte.<br>Based on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose two different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first model treats the diffusion-controlled regime of interface motion, while the second model is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. These models generalise the well-known Mullins-Sekerka model for Ostwald ripening. We concentrate on arsenic-rich liquid spherical droplets in a gallium arsenide crystal. Droplets can shrink or grow with time but the centres of droplets remain fixed. The liquid is assumed to be homogeneous in space. Due to different scales for typical distances between droplets and typical radii of liquid droplets we can derive formally so-called mean field models. For a model in the diffusion-controlled regime we prove this limit by homogenisation techniques under plausible assumptions. These mean field models generalise the Lifshitz-Slyozov-Wagner model, which can be derived from the Mullins-Sekerka model rigorously, see Niethammer et al., and is well-understood. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. We determine possible equilibria and discuss their stability. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.

Les modèles de milieux poreux, en régime compressible ou incompressible, sont utilisés dans la littérature pour décrire les propriétés mécaniques des tissus vivants et en particulier de la croissance tumorale. Il est possible de construire un lien entre ces deux différentes représentations en utilisant une loi de pression raide. Dans la limite incompressible, les modèles compressibles conduisent à des problèmes de frontières libres de type Hele-Shaw. Nos travaux visent à étudier la limite de pression raide des équations de type milieu poreux motivées par le développement tumoral. Notre première étude concerne l’analyse et la simulation numérique d’un modèle incluant l’effet des nutriments. Ensuite, un système d’équations, dont le couplage est délicat, décrit la densité cellulaire et la concentration en nutriments. Pour cette raison, la dérivation de l’équation de pression dans la limite incompressible était un problème ouvert qui nécessite la compacité forte du gradient de pression. Pour l’établir, nous utilisons deux nouvelles idées : une version L3 de la célèbre estimation d’Aronson-Bénilan, également utilisée récemment pour des problèmes connexes, et une estimation L4 sur le gradient de pression (où l’exposant 4 est optimal). Nous étudions en outre l’optimalité de cette estimation par un schéma numérique upwind aux différences finies, que nous montrons être stable et asymptotic preserving. Notre deuxième étude est centrée sur l’équation de milieux poreux avec effets convectifs. Nous étendons les techniques développées pour le cas avec nutriments, trouvant ainsi la relation de complémentarité sur la pression limite. De plus, nous fournissons une estimation du taux de convergence à la limite incompressible. Enfin, nous étudions un système multi-espèces. En particulier, en tenant compte de l’hétérogénéité phénotypique, nous incluons une variable structurée dans le problème. Par conséquent, un système de diffusion croisée et dégénérée décrit l’évolution des distributions phénotypiques. En adaptant des méthodes récemment développées pour des systèmes à deux équations, nous prouvons l’existence de solutions faibles et nous passons à la limite incompressible. En outre, nous prouvons de nouveaux résultats de régularité sur la pression totale, qui est liée à la densité totale par une loi de puissance<br>Both compressible and incompressible porous medium models have been used in the literature to describe the mechanical aspects of living tissues, and in particular of tumor growth. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems of Hele-Shaw type where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. Then, a coupled system of equations describes the cell density and the nutrient concentration. For this reason, the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, also recently applied to related problems, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state

In this thesis, we study mathematical models that describe the morphology of a generalized biological cell in equilibrium or under the influence of external forces. Within these models, the cell is considered as a thermodynamic system, where streaming effects in the cell bulk and the surrounding are coupled with a Helfrich-type model for the cell membrane. The governing evolution equations for the cell given in a continuum formulation are derived using an energy variation approach. Such two-phase flow problems that combine streaming effects with a free boundary problem that accounts for bending and surface tension can be described effectively by a diffuse interface approach. An advantage of the diffuse interface approach is that models for e.g. different biophysical processes can easily be combined. That makes this method suitable to describe complex phenomena such as cell motility and multi-cell dynamics. Within the first model for cell motility, we combine a biological network for GTPases with the hydrodynamic Helfrich-type model. This model allows to account for cell motility driven by membrane protrusion as a result of actin polymerization. Within the second model, we moreover extend the Helfrich-type model by an active gel theory to account for the actin filaments in the cell bulk. Caused by contractile stress within the actin-myosin solution, a spontaneous symmetry breaking event occurs that lead to cell motility. In this thesis, we further study the dynamics of multiple cells which is of wide interest since it reveals rich non-linear behavior. To apply the diffuse interface framework, we introduce several phase field variables to account for several cells that are coupled by a local interaction potential. In a first application, we study white blood cell margination, a biological phenomenon that results from the complex relation between collisions, different mechanical properties and lift forces of red blood cells and white blood cells within the vascular system. Here, it is shown that inertial effects, which can become of relevance in various parts of the cardiovascular system, lead to a decreasing tendency for margination with increasing Reynolds number. Finally, we combine the active polar gel theory and the multi-cell approach that is capable of studying collective migration of cells. This hydrodynamic approach predicts that collective migration emerges spontaneously forming coherently-moving clusters as a result of the mutual alignment of the velocity vectors during inelastic collisions. We further observe that hydrodynamics heavily influence those systems. However, a complete suppression of the onset of collective migration cannot be confirmed. Moreover, we give a brief insight how such highly coupled systems can be treated numerically using finite elements and how the numerical costs can be limited using operator splitting approaches and problem parallelization with OPENMP<br>Diese Dissertation beschäftigt sich mit mathematischen Modellen zur Beschreibung von Gleichgewichts- und dynamischen Zuständen von verallgemeinerten biologischen Zellen. Die Zellen werden dabei als thermodynamisches System aufgefasst, bei dem Strömungseffekte innerhalb und außerhalb der Zelle zusammen mit einem Helfrich-Modell für Zellmembranen kombiniert werden. Schließlich werden durch einen Energie-Variations-Ansatz die Evolutionsgleichungen für die Zelle hergeleitet. Es ergeben sie dabei Mehrphasen-Systeme, die Strömungseffekte mit einem freien Randwertproblem, das zusätzlich physikalischen Einflüssen wie Biegung und Oberflächenspannung unterliegt, vereinen. Um solche Probleme effizient zu lösen, wird in dieser Arbeit die Diffuse-Interface-Methode verwendet. Ein Vorteil dieser Methode ist, dass es sehr einfach möglich ist, Modelle, die verschiedenste Prozesse beschreiben, miteinander zu vereinen. Dies erlaubt es, komplexe biologische Phänomene, wie zum Beispiel Zellmotilität oder auch die kollektive Bewegung von Zellen, zu beschreiben. In den Modellen für Zellmotilität wird ein biologisches Netzwerk-Modell für GTPasen oder auch ein Active-Polar-Gel-Modell, das die Aktinfilamente im Inneren der Zellen als Flüssigkristall auffasst, mit dem Multi-Phasen-Modell kombiniert. Beide Modelle erlauben es, komplexe Vorgänge bei der selbst hervorgerufenen Bewegung von Zellen, wie das Vorantreiben der Zellmembran durch Aktinpolymerisierung oder auch die Kontraktionsbewegung des Zellkörpers durch kontraktile Spannungen innerhalb des Zytoskelets der Zelle, zu verstehen. Weiterhin ist die kollektive Bewegung von vielen Zellen von großem Interesse, da sich hier viele nichtlineare Phänomene zeigen. Um das Diffuse-Interface-Modell für eine Zelle auf die Beschreibung mehrerer Zellen zu übertragen, werden mehrere Phasenfelder eingeführt, die die Zellen jeweils kennzeichnen. Schließlich werden die Zellen durch ein lokales Abstoßungspotential gekoppelt. Das Modell wird angewendet, um White blood cell margination, das die Annäherung von Leukozyten an die Blutgefäßwand bezeichnet, zu verstehen. Dieser Prozess wird dabei bestimmt durch den komplexen Zusammenhang zwischen Kollisionen, den jeweiligen mechanischen Eigenschaften der Zellen, sowie deren Auftriebskraft innerhalb der Adern. Die Simulationen zeigen, dass diese Annäherung sich in bestimmten Gebieten des kardiovaskulären Systems stark vermindert, in denen die Blutströmung das Stokes-Regime verlässt. Schließlich wird das Active-Polar-Gel-Modell mit dem Modell für die kollektive Bewegung vom Zellen kombiniert. Dies macht es möglich, die kollektive Bewegung der Zellen und den Einfluss von Hydrodynamik auf diese Bewegung zu untersuchen. Es zeigt sich dabei, dass der Zustand der kollektiven gerichteten Bewegung sich spontan aus der Neuausrichtung der jeweiligen Zellen durch inelastische Kollisionen ergibt. Obwohl die Hydrodynamik einen großen Einfluss auf solche Systeme hat, deuten die Simulationen nicht daraufhin, dass Hydrodynamik die kollektive Bewegung vollständig unterdrückt. Weiterhin wird in dieser Arbeit gezeigt, wie die stark gekoppelten Systeme numerisch gelöst werden können mit Hilfe der Finiten-Elemente-Methode und wie die Effizienz der Methode gesteigert werden kann durch die Anwendung von Operator-Splitting-Techniken und Problemparallelisierung mittels OPENMP

Reservoir testing and analysis are fundamental tools in understanding reservoir hydraulics and hence forecasting reservoir responses. The quality of the analysis is very dependent on the conceptual model used in investigating the responses under different flowing conditions. The use of reservoir testing in the characterization and derivation of reservoir parameters is widely established, especially in conventional oil and gas reservoirs. However, with depleting conventional reserves, the quest for unconventional reservoirs to secure the increasing demand for energy is increasing; which has triggered intensive research in the fields of reservoir characterization. Gas hydrate reservoirs, being one of the unconventional gas reservoirs with huge energy potential, is still in the juvenile stage with reservoir testing as compared to the other unconventional reservoirs. The endothermic dissociation hydrates to gas and water requires addressing multiphase flow and heat energy balance, which has made efforts to develop reservoir testing models in this field difficult. As of now, analytically quantifying the effect on hydrate dissociation on rate and pressure transient responses are till date a huge challenge. During depressurization, the heat energy stored in the reservoir is used up and due to the endothermic nature of the dissociation; heat flux begins from the confining layers. For Class 3 gas hydrates, just heat conduction would be responsible for the heat influx and further hydrate dissociation; however, the moving boundary problem could also be an issue to address in this reservoir, depending on the equilibrium pressure. To address heat flux problem, a proper definition of the inner boundary condition for temperature propagation using a Clausius-Clapeyron type hydrate equilibrium model is required. In Class 1 and 2, crossflow problems would occur and depending on the layer of production, convective heat influx from the free fluid layer and heat conduction from the cap rock of the hydrate layer would be further issues to address. All these phenomena make the derivation of a suitable reservoir testing model very complex. However, with a strong combination of heat energy and mass balance techniques, a representative diffusivity equation can be derived. Reservoir testing models have been developed and responses investigated for different boundary conditions in normally pressured Class 3 gas hydrates, over-pressured Class 3 gas hydrates (moving boundary problem) and Class 1 and 2 gas hydrates (crossflow problem). The effects of heat flux on the reservoir responses have been addressed in detail.

abstract: In a 2004 paper, John Nagy raised the possibility of the existence of a hypertumor \emph{i.e.}, a focus of aggressively reproducing parenchyma cells that invade part or all of a tumor. His model used a system of nonlinear ordinary differential equations to find a suitable set of conditions for which these hypertumors exist. Here that model is expanded by transforming it into a system of nonlinear partial differential equations with diffusion, advection, and a free boundary condition to represent a radially symmetric tumor growth. Two strains of parenchymal cells are incorporated; one forming almost the entirety of the tumor while the much more aggressive strain appears in a smaller region inside of the tumor. Simulations show that if the aggressive strain focuses its efforts on proliferating and does not contribute to angiogenesis signaling when in a hypoxic state, a hypertumor will form. More importantly, this resultant aggressive tumor is paradoxically prone to extinction and hypothesize is the cause of necrosis in many vascularized tumors.<br>Dissertation/Thesis<br>Doctoral Dissertation Applied Mathematics 2014

A two-dimensional Boundary Element Method (BEM) is developed to study the strongly nonlinear interaction between a surface-piercing body and the free-surface. The scheme is applied to problems with and without the possibility of ventilation resulting from the motion and geometric configuration of the surface-piercing body. The main emphasis of this research work is on the development of numerical methods to improve the performance prediction of surface-piercing propellers by including the whole range of free-surface nonlinearities. The scheme is applied to predict the ventilated cavity shapes resulting from the vertical and rotational motion of a blade-section with fully nonlinear free-surface boundary conditions. The current method is able to predict the ventilated cavity shapes for a wide range of angles of attack and Froude numbers, and is in good agreement with existing experimental results. Through a comparison with a linearized free-surface method, the current method highlights the shortcomings of the negative image approach used commonly in two-dimensional and three-dimensional numerical methods for surface-piercing hydrofoils or propellers. The current method with all its capabilities makes it a unique contribution to improving numerical tools for the performance prediction of surface-piercing propellers. The scheme is also applied to predict the roll and heave dynamics of two-dimensional Floating Production Storage and Offloading (FPSO) vessel hull sections within a potential flow framework. The development of the potential flow model is aimed at validating the free-surface dynamics of an independently developed Navier Stokes Solver for predicting the roll characteristics of two-dimensional hull sections with bilge keels.<br>text

Zhao, Xiaozheng = 滑移邊界條件下湍流熱對流的實驗研究 / 趙晓争.<br>"September 2010."<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.<br>Includes bibliographical references (leaves 52-57).<br>Abstracts in English and Chinese.<br>Zhao, Xiaozheng = Hua yi bian jie tiao jian xia tuan liu re dui liu de shi yan yan jiu / Zhao Xiaozheng.<br>Abstract --- p.i<br>摘要 --- p.ii<br>Acknowledgement --- p.iv<br>Contains --- p.iv<br>List of Figures --- p.vii<br>List of Tables --- p.xi<br>Chapter 1 --- Introduction --- p.1<br>Chapter 1.1 --- Turbulence --- p.1<br>Chapter 1.2 --- Turbulent Rayleigh-Benard Convection --- p.2<br>Chapter 1.2.1 --- Physical Picture --- p.2<br>Chapter 1.2.2 --- Governing Equations and Characteristic Parameters --- p.5<br>Chapter 1.2.3 --- Nu Scaling --- p.7<br>Chapter 1.2.4 --- Boundary Layer --- p.8<br>Chapter 1.3 --- Motivations of the Present Work --- p.10<br>Chapter 2 --- Experimental Setup --- p.13<br>Chapter 2.1 --- The Convection Cell --- p.13<br>Chapter 2.2 --- Temperature Probe and Translation Stage --- p.15<br>Chapter 2.3 --- Calibration of the Thermistors --- p.17<br>Chapter 2.4 --- Data Acquisition Units --- p.18<br>Chapter 2.5 --- The Working Fluids --- p.19<br>Chapter 2.6 --- Heat Leakage Prevention --- p.21<br>Chapter 3 --- Heat Transfer and Thermal Boundary Layer Measurement --- p.23<br>Chapter 3.1 --- The Setup and Experimental Procedure --- p.23<br>Chapter 3.2 --- The Mean Temperature and Temperature Fluctuation Profiles across the Interfaces --- p.24<br>Chapter 3.2.1 --- Profiles across the Water-FC77 Interface --- p.24<br>Chapter 3.2.2 --- Profiles across the FC77-Mercury Interface --- p.27<br>Chapter 3.3 --- Nu Results --- p.29<br>Chapter 3.3.1 --- Results Obtained with Assumption of Pure Conduction --- p.30<br>Chapter 3.3.2 --- Results from Mean Temperature Profile --- p.32<br>Chapter 3.3.3 --- Comparison of the Two Methods --- p.33<br>Chapter 3.4 --- Boundary Layer Thickness --- p.37<br>Chapter 3.5 --- Summary --- p.39<br>Chapter 4 --- Influence of Flow in the Water (Mercury) Layer on the FC77 Layer --- p.41<br>Chapter 4.1 --- Experimental Setup --- p.41<br>Chapter 4.2 --- Main Results --- p.42<br>Chapter 4.3 --- Probability Density Function and Temperature Oscillation --- p.44<br>Chapter 4.4 --- Summary --- p.50<br>Chapter 5 --- Conclusions and Perspective --- p.51<br>Chapter 5.1 --- Conclusions --- p.51<br>Chapter 5.2 --- Perspective for Future Work --- p.52

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems.<br>For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function.<br>For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods.<br>本論文為金融衍生產品定價的兩類問題作出了研究。第一類是自由邊界問題，它可以制定一個受制於自由邊界條件的偏微分方程式(PDE)，雖然當中自由邊界條件的函數形式是已知的，但自由邊界的位置是未知的，只能通過為實際解施加連續性條件作隱式確定。這裡為兩個具體問題進行了研究，分別是固定利率按揭合約(fixed-rate mortgages)定價和方差恆彈性模型的美式期權(CEV American options)定價。第二類是多維問題，它涉及到多個相關隨機變量及他們引申出的多維PDE。這裡為一個典型例子進行了研究，稱為籃子差異期權(basket-spread options)，其基礎資產價格由相關的幾何布朗運動驅動。我們為這三個問題提出了解析近似解。<br>對於上述的自由邊界問題，我們提出了一項參數移動邊界來近似模仿未知的自由邊界，使原來的自由邊界問題轉化為移動邊界問題，從而提出一種解析近似解。控制移動邊界的參數是通過滿足近似解的一階導數連續性條件來定。得到了解析近似解令當中的衍生產品定價和避險參數能有效快速地計算出，相比於有限差分法(finite-difference method)，精度保持了但計算時間顯著降低。再透過應用一個多階段方案，將移動邊界重鑄成一項分段光滑的連續函數，能有系統地將近似解的結果逼近有限差分法的結果。<br>對於上述的多維問題，我們從Kirk(1995)的二維差異期權(spread option)近似解定價公式推廣到多維的籃子差異期權。由於最終的定價公式是封閉形式，所有避險參數也從而得到封閉式近似解。從一些模擬例子顯示出，近似解的定價和避險參數，與通過數值積分法(numerical integration)或蒙地卡羅模擬法(Monte Carlo simulation)獲得的基準值比較，只有小於百分之一的誤差。此外，透過利用一種期權價格和相關基礎變量的概率分佈關係，我們進一步推論出一項籃子差異變量的近似解分佈函數，這可應用到任何多維幾何布朗運動的線性組合變量分佈。最後，我們提出一種隱式攝動方法，把定價誤差減少高達一百倍，跟現有的近似解定價方法相比，這是其中一種最健全和準確的籃子差異期權定價方法。<br>Lau, Chun Sing = 自由邊界和多維的金融衍生產品定價問題 : 解析近似解 / 劉振聲.<br>Thesis Ph.D. Chinese University of Hong Kong 2014.<br>Includes bibliographical references (leaves 174-186).<br>Abstracts also in Chinese.<br>Title from PDF title page (viewed on 12, September, 2016).<br>Lau, Chun Sing = Zi you bian jie he duo wei de jin rong yan sheng chan pin ding jia wen ti : jie xi jin si jie / Liu Zhensheng.<br>Detailed summary in vernacular field only.<br>Detailed summary in vernacular field only.<br>Detailed summary in vernacular field only.

Investigation of material behaviour in a squeeze flow geometry provides an impor- tant technique in rheology and it is relevant also from the technological point of view (some types of dampers, compression moulding). To our best knowledge, the sque- eze flow has not been solved for fluids-like materials with pressure-dependent material moduli. In the main scope of the present thesis, an incompressible fluid whose visco- sity strongly depends on the pressure is studied in both the perfect-slip and the no-slip squeeze flow. It is shown that such a material model can provide interesting departures compared to the classical model for viscous (Navier-Stokes) fluid even on the level of analytical solutions, which are obtained using some physically relevant simplificati- ons. Numerical simulation of a free boundary problem for the no-slip squeeze flow is then developed in the thesis using body-fitted curvilinear coordinates and spectral collocation method. An interesting behaviour is expected especially in the corners of the computational domain where the stress singularities are normally located. Unfor- tunately, numerical results reveal some fundamental drawbacks related to the physical model and its possible improvement is discussed at the end of the thesis.

Reservoir testing and analysis are fundamental tools in understanding reservoir hydraulics and hence forecasting reservoir responses. The quality of the analysis is very dependent on the conceptual model used in investigating the responses under different flowing conditions. The use of reservoir testing in the characterization and derivation of reservoir parameters is widely established, especially in conventional oil and gas reservoirs. However, with depleting conventional reserves, the quest for unconventional reservoirs to secure the increasing demand for energy is increasing; which has triggered intensive research in the fields of reservoir characterization. Gas hydrate reservoirs, being one of the unconventional gas reservoirs with huge energy potential, is still in the juvenile stage with reservoir testing as compared to the other unconventional reservoirs. The endothermic dissociation hydrates to gas and water requires addressing multiphase flow and heat energy balance, which has made efforts to develop reservoir testing models in this field difficult. As of now, analytically quantifying the effect on hydrate dissociation on rate and pressure transient responses are till date a huge challenge. During depressurization, the heat energy stored in the reservoir is used up and due to the endothermic nature of the dissociation; heat flux begins from the confining layers. For Class 3 gas hydrates, just heat conduction would be responsible for the heat influx and further hydrate dissociation; however, the moving boundary problem could also be an issue to address in this reservoir, depending on the equilibrium pressure. To address heat flux problem, a proper definition of the inner boundary condition for temperature propagation using a Clausius-Clapeyron type hydrate equilibrium model is required. In Class 1 and 2, crossflow problems would occur and depending on the layer of production, convective heat influx from the free fluid layer and heat conduction from the cap rock of the hydrate layer would be further issues to address. All these phenomena make the derivation of a suitable reservoir testing model very complex. However, with a strong combination of heat energy and mass balance techniques, a representative diffusivity equation can be derived. Reservoir testing models have been developed and responses investigated for different boundary conditions in normally pressured Class 3 gas hydrates, over-pressured Class 3 gas hydrates (moving boundary problem) and Class 1 and 2 gas hydrates (crossflow problem). The effects of heat flux on the reservoir responses have been addressed in detail.