本論文主要目標為發展金融互換的定價模型。既是欲建立量化模型，首要 工作在於對量化對象 --- 金融互換工具的實際特性， 實務上的運作，有 瞭解與掌握，再輔以必要的數量基礎，方不致於`` 失真 '' 。本文共分 五章首章為緒論，第二章為對金融互換工具的全盤認識，試圖由金融互換 的契約切入，進而歸納分類要件，演化及最終種類，最後提出定價時的幾 個思維面向(Dimensions )。第三章為文獻回顧，指出金融互換定價模型 的基礎，為建立在具浮動利率金融工具的定價模型上。 第四章為發展理 論模型基礎及數值分析結果。第五章為結論。

The goal of this thesis in finance is to combine the use of advanced mathematical methods with a return to foundational economic issues. In that perspective, I study generalized rational expectations and asset pricing in Chapter 2, and a converse comparison principle for backward stochastic differential equations with jumps in Chapter 3. Since the use of stochastic methods in finance is an interesting and complex issue in itself - if only to clarify the difference between the use of mathematical models in finance and in physics or biology - I also present a philosophical reflection on the interpretation of mathematical models in finance (Chapter 4). In Chapter 5, I conclude the thesis with an essay on the history and interpretation of mathematical probability - to be read while keeping in mind the fundamental role of mathematical probability in financial models.

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70. / QC 20100825

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles. In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem. The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’). The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion. In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.

Le cliquet de Muller est un modèle mathématiques illustrant l'accumulation de mutations délétères dans une population asexuée. L'idée principale est que l'absence de recombinaison oblige les enfants à avoir au moins autant de mutations nocives que leurs parents, et au bout d'un certain temps, le nombre minimum de mutations délétères de la population, qui est donc un processus croissant, augmente : on dit alors que le cliquet clique. Le modèle du cliquet de Muller qui est étudié dans cette thèse est un système infini d'équations différentielles stochastiques de Fleming-Viot couplées. On montre dans une première partie d'abord que le cliquet s'actionne en temps fini p.s., puis que l'espérance du temps mis pour cliquer est également finie. On utilise pour cela des comparaisons d'équations stochastiques et des changements de temps. Dans une deuxième partie, on démontre que ce modèle est équivalent à un modèle du look-down modifié auquel on a ajouté des mutations et des morts. Puis dans la troisième partie on généralise le résultat de la deuxième à un cadre plus large de systèmes d'équations différentielles stochastiques. / Muller's Ratchet is a model from evolutionary theory describing the accumulation of deleterious mutations in asexually reproducing population. The lack of recombination implies that children have all the deleterious mutations of his parent. The minimal number of deleterious mutations carried in the population is an non-decreasing process, and if it increases we say that the Muller's ratchet clicks. The model studied in this thesis is an infinite system of stochastic differential equations. In the first chapter, we first prove that the ratchet clicks in finite time a.s., then that the clicking time has finite expectation. For this we use comparison arguments and time changes. In the second chapter, we prove that this model is equivalent to a modified look-down model with mutation and selection. In the third chapter we generalize the results of chapter 2 to a more general model.

Submitted by Pablo Aguiar De Maio (pabloamaio@outlook.com) on 2015-09-10T19:50:43Z No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2015-09-25T12:16:10Z (GMT) No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2015-09-28T16:51:14Z (GMT) No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Made available in DSpace on 2015-09-28T16:51:50Z (GMT). No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) Previous issue date: 2015-07-31 / In this work we present a new numerical method with adaptive stepsize based on the local linearization approach, to integrate stochastic differential equations with additive noise. We also propose a computational scheme that allows efficient implementation of this method, properly adapting the algorithm of Padé with scaling-squaring strategy to compute the exponential of matrices involved. To introduce the construction of this method, we briefly explain what stochastic differential equations are, the mathematics that is behind them, their relevance to the modeling of various phenomena, and the importance of using numerical methods to evaluate this kind of equations. A succinct study of numerical stability is also presented on the following pages. With this dissertation, we intend to introduce the necessary basis for the construction of the new method/scheme. At the end, several numerical experiments are performed to demonstrate, in a practical way, the effectiveness of the proposed method, comparing it with other methods commonly used. / Neste trabalho apresentamos um novo método numérico com passo adaptativo baseado na abordagem de linearização local, para a integração de equações diferenciais estocásticas com ruído aditivo. Propomos, também, um esquema computacional que permite a implementação eficiente deste método, adaptando adequadamente o algorítimo de Padé com a estratégia “scaling-squaring” para o cálculo das exponenciais de matrizes envolvidas. Antes de introduzirmos a construção deste método, apresentaremos de forma breve o que são equações diferenciais estocásticas, a matemática que as fundamenta, a sua relevância para a modelagem dos mais diversos fenômenos, e a importância da utilização de métodos numéricos para avaliar tais equações. Também é feito um breve estudo sobre estabilidade numérica. Com isto, pretendemos introduzir as bases necessárias para a construção do novo método/esquema. Ao final, vários experimentos numéricos são realizados para mostrar, de forma prática, a eficácia do método proposto, e compará-lo com outros métodos usualmente utilizados.

L'objectif de notre thèse de <p>doctorat est d’étudier et de décrire les propriétés chimiques et mé- <p>caniques du moteur moléculaire F1 -ATPase. Le moteur F1 -ATPase <p>est un moteur rotatif, d’aspect sphérique et d’environ 10 nanomètre <p>de rayon, qui utilise l’énergie de l’hydrolyse de l’ATP comme car- <p>burant moléculaire. <p>Des questions fondamentales se posent sur le fonctionnement de <p>ce moteurs et sur la quantité de travail qu’il peut fournir. Il s’agit <p>de questions qui concernent principalement la thermodynamique <p>des processus irréversibles. De plus, comme ce moteur est de <p>taille nanométrique, il est fortement influencé par les fluctuations <p>moléculaires, ce qui nécessite une approche stochastique. <p>C’est en créant deux modéles stochastiques complémentaires de <p>ce moteur que nous avons contribué à répondre à ces questions <p>fondamentales. <p>Le premier modèle discuté au chapitre 5 de la thèse, est un mod- <p>èle continu dans le temps et l’espace, décrit par des équations de <p>Fokker-Planck, est construit sur des résultats expérimentaux. <p>Ce modèle tient compte d’une description explicite des fluctua- <p>tions affectant le degré de liberté mécanique et décrit les tran- <p>sitions entre les différents états chimiques discrets du moteur, <p>par un processus de sauts aléatoires entre premiers voisins. Nous <p>avons obtenus des résultats précis concernant la chimie d’hydrolyse <p>et de synthèse de l’ATP, et pour les dépendences du moteur en les <p>différentes variables mécaniques, à savoir, la friction et le couple <p>de force extérieur, ainsi que la dépendence en la température. <p>Les résultats que nous avons obtenus avec ce modèle sont en ex- <p>cellent accord avec les observations expérimentales. <p>Le second modèle est discret dans l’espace et continu dans le <p>temps et est décrit dans le chapitre 6. L’analyse des résultats <p>obtenus par simulations numériques montre que le modèle est <p>en accord avec les observations expérimentales et il permet en <p>outre de dériver des grandeurs thermodynamiques analytique- <p>ment, décrites au chapitre 4, ce que le modèle continu ne permet <p>pas. <p>La comparaison des deux modèles révele la nature du fonction- <p>nement du moteur, ainsi que son régime de fonctionnement loin <p>de l’équilibre. Le second modèle a éte soumis récemment pour <p>publication. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

L’objectif central de la thèse est d’étudier diverses mesures du risque de modèle, exprimées en terme monétaire, qui puissent être appliquées de façon cohérente à une collection hétérogène de produits financiers. Les deux premiers chapitres traitent cette problématique, premièrement d’un point de vue théorique, ensuite en menant un étude empirique centrée sur le marché du gaz naturel. Le troisième chapitre se concentre sur une étude théorique du risque dit de base (en anglais basis risk). Dans le premier chapitre, nous nous sommes intéressés à l’évaluation de produits financiers complexes, qui prend en compte le risque de modèle et la disponibilité dans le marché de produits dérivés basiques, appelés aussi vanille. Nous avons en particulier poursuivi l’approche du transport optimal (connue dans la littérature) pour le calcul des bornes de prix et des stratégies de sur (sous)-couverture robustes au risque de modèle. Nous reprenons en particulier une construction de probabilités martingales sous lesquelles le prix d’une option exotique atteint les dites bornes de prix, en se concentrant sur le cas des martingales positives. Nous mettons aussi en évidence des propriétés significatives de symétrie dans l’étude de ce problème. Dans le deuxième chapitre, nous approchons le problème du risque de modèle d’un point de vue empirique, en étudiant la gestion optimale d’une unité de gaz naturel et en quantifiant l’effet de ce risque sur sa valeur optimale. Lors de cette étude, l’évaluation de l’unité de stockage est basée sur le prix spot, alors que sa couverture est réalisée avec des contrats à termes. Comme mentionné auparavant, le troisième chapitre met l’accent sur le risque de base, qui intervient lorsque l’on veut couvrir un actif conditionnel basé sur un actif non traité (par exemple la température) en se servant d’un portefeuille constitué d’actifs traités sur le marché. Un critère de couverture dans ce contexte est celui de la minimisation de la variance qui est étroitement lié à la décomposition dite de Föllmer-Schweizer. Cette décomposition peut être déduite de la résolution d’une certaine équation différentielle stochastique rétrograde (EDSR) dirigée par une martingale éventuellement à sauts. Lorsque cette martingale est un mouvement brownien standard, les EDSR sont fortement associées aux EDP paraboliques semi linéaires. Dans le cas général nous formulons un problème déterministe qui étend les EDPs mentionnées. Nous appliquons cette démarche à l’important cas particulier de la décomposition de Föllmer-Schweizer, dont nous donnons des expressions explicites de la décomposition du payoff d’une option lorsque les sous-jacents sont exponentielles de processus additifs. / The main objective of this thesis is the study of the model risk and its quantification through monetary measures. On the other hand we expect it to fit a large set of complex (exotic) financial products. The first two chapters treat the model risk problem both from the empirical and the theoretical point of view, while the third chapter concentrates on a theoretical study of another financial risk called basis risk. In the first chapter of this thesis, we are interested in the model-independent pricing and hedging of complex financial products, when a set of standard (vanilla) products are available in the market. We follow the optimal transport approach for the computation of the option bounds and the super (sub)-hedging strategies. We characterize the optimal martingale probability measures, under which the exotic option price attains the model-free bounds; we devote special interest to the case when the martingales are positive. We stress in particular on the symmetry relations that arise when studying the option bounds. In the second chapter, we approach the model risk problem from an empirical point of view. We study the optimal management of a natural gas storage and we quantify the impact of that risk on the gas storage value. As already mentioned, the last chapter concentrates on the basis risk, which is the risk that arises when one hedges a contingent claim written on a non-tradable but observable asset (e.g. the temperature) using a portfolio of correlated tradable assets. One hedging criterion is the mean-variance minimization, which is closely related to the celebrated Föllmer-Schweizer decomposition. That decomposition can be deduced from the resolution of a special Backward Stochastic Differential Equations (BSDEs) driven by a càdlàg martingale. When this martingale is a standard Brownian motion, the related BSDEs are strongly related to semi-linear parabolic PDEs. In that chapter, we formulate a deterministic problem generalizing those PDEs to the general context of martingales and we apply this methodology to discuss some properties of the Föllmer-Schweizer decomposition. We also give an explicit expression of such decomposition of the option payoff when the underlying prices are exponential of additives processes.