From both theoretical and applied perspectives, first passage<br />time problems for random processes are challenging and of great<br />interest. In this thesis, our contribution consists on providing<br />explicit or quasi-explicit solutions for these problems in two<br />different settings.<br /><br />In the first one, we deal with problems related to the<br />distribution of the first passage time (FPT) of a Brownian motion<br />over a continuous curve. We provide several representations for<br />the density of the FPT of a fixed level by an Ornstein-Uhlenbeck<br />process. This problem is known to be closely connected to the one<br />of the FPT of a Brownian motion over the square root boundary.<br />Then, we compute the joint Laplace transform of the $L^1$ and<br />$L^2$ norms of the $3$-dimensional Bessel bridges. This result is<br />used to illustrate a relationship which we establish between the<br />laws of the FPT of a Brownian motion over a twice continuously<br />differentiable curve and the quadratic and linear ones. Finally,<br />we introduce a transformation which maps a continuous function<br />into a family of continuous functions and we establish its<br />analytical and algebraic properties. We deduce a simple and<br />explicit relationship between the densities of the FPT over each<br />element of this family by a selfsimilar diffusion.<br /><br /> In the second setting, we are concerned with the study of<br />exit problems associated to Generalized Ornstein-Uhlenbeck<br />processes. These are constructed from the classical<br />Ornstein-Uhlenbeck process by simply replacing the driving<br />Brownian motion by a Lévy process. They are diffusions with<br />possible jumps. We consider two cases: The spectrally negative<br />case, that is when the process has only downward jumps and the<br />case when the Lévy process is a compound Poisson process with<br />exponentially distributed jumps. We derive an expression, in terms<br />of new special functions, for the joint Laplace transform of the<br />FPT of a fixed level and the primitives of theses processes taken<br />at this stopping time. This result allows to compute the Laplace<br />transform of the price of a European call option on the maximum on<br />the yield in the generalized Vasicek model. Finally, we study the<br />resolvent density of these processes when the Lévy process is<br />$\alpha$-stable ($1 < \alpha \leq 2$). In particular, we<br />construct their $q$-scale function which generalizes the<br />Mittag-Leffler function.

Notre travail porte sur le phénomène de cutoff pour des n-échantillons de processus de Markov, dans le but de l'appliquer à la détection de la convergence d'algorithmes parallélisés. Dans un premier temps, le processus échantillonné est un processus d'Ornstein-Uhlenbeck. Nous mettons en évidence le phénomène de cutoff pour le n-échantillon, puis nous faisons le lien avec la convergence en loi du temps d'atteinte par le processus moyen d'un niveau fixé. Dans un second temps, nous traitons le cas général où le processus échantillonné converge à vitesse exponentielle vers sa loi stationnaire. Nous donnons des estimations précises des distances entre la loi du n-échantillon et sa loi stationnaire. Enfin, nous expliquons comment aborder les problèmes de temps d'atteinte liés au phénomène du cutoff.

Les dérivés de crédit ont connu en quelques années un développement fulgurant en finance : les volumes de transactions ont augmenté exponentiellement, de nouveaux produits ont été créés. La récente crise du sub-prime a mis en évidence l'insuffisance des modèles actuels. Le but de cette thèse est de créer de nouveaux modèles mathématiques qui prennent en compte la dynamique de dépendance (« tail dependence ») des marchés. Après une revue de la littérature et des modèles existants, nous nous focalisons sur la modélisation de la « corrélation » (ou plus exactement la dynamique de la structure de dépendance) entre différentes entités dans un portefeuille de crédit (CDO). Dans une première phase, une formulation simple des copules dynamiques est proposée. Ensuite, nous présentons une seconde formulation en utilisant des processus de Lévy à spectre positif (i.e. gamma Ornstein-Uhlenbeck). L'écriture de cette nouvelle famille de copules archimédiennes nous permet d'obtenir une formule asymptotique simple pour la distribution des pertes d'un portefeuille de crédit granulaire. L'une des particularités du modèle proposé est sa capacité de reproduire des dépendances extrêmes comparables aux phénomènes récents de contagion sur les marchés comme la crise du « subprime » aux Etats-Unis. Une application sur l'estimation des prix des tranches de CDOs est aussi présentée. Dans cette thèse, nous proposons également d'utiliser des copules dynamiques pour modéliser des migrations jointes des qualités de crédit afin de prendre en compte les co-migrations extrêmes. En effet, les copules nous permettent d'étendre notre connaissance des processus de migration mono-variable à un cadre multi-variables. Afin de tenir compte de multiples sources de risques systémiques, nous développons des copules dynamiques à plusieurs facteurs. Enfin, nous montrons que la brique élémentaire de structure de dépendance induite par une mesure du temps aléatoire « Time Changed Process » rentre dans le cadre des copules dynamiques.

Ce travail se place dans le cadre de la statistique infinidimensionnelle . Par généralisation en dimension quelconque de certaines méthodes d'analyse multidimensionnelle classique il fournit des solutions satisfaisantes pour des problèmes de décision concernant la moyenne de certains processus gaussiens.<br /><br />La première partie est consacrée à l'étude de tests<br />quadratiques d' hypothèses linéaires et à l'extension en dimension infinie du modèle I d' analyse de la variance.<br /><br />Dans la deuxième partie - les aspects probabilistes d'un modèle mathématique pour la réponse en potentiel d'un neurone sont étudiés et une application de l'analyse de la variance est développée.<br /><br />Enfin le dernier chapitre aborde les problèmes de calcul effectif des régions critiques des tests utilisés .

In this paper, we study mathematical properties of a generalized bivariate Ornstein-Uhlenbeck model for financial assets. Originally introduced by Lo and Wang, this model possesses a stochastic drift term which influences the statistical properties of the asset in the real (observable) world. Furthermore, we generali- ze the model with respect to a time-dependent (but still non-random) volatility function. Although it is well-known, that drift terms - under weak regularity conditions - do not affect the behaviour of the asset in the risk-neutral world and consequently the Black-Scholes option pricing formula holds true, it makes sense to point out that these regularity conditions are fulfilled in the present model and that option pricing can be treated in analogy to the Black-Scholes case.

Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems. The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution. Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations. In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely. Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.

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本論文我們延伸Goldberg(1986)之結論,採用Ornstein Uhlenbeck positon process取代一般幾何布朗尼運動之假設來評價選擇權.Goldberg(1986)認為Ornstein Uhlenbeck positon process比幾何布朗尼運動更適合用來描述在不完全市場下之股價波動過程.我們在此波動過程的假設下,推倒出在風險中立的機率測度下歐式選擇權的評價模型及其避險參數,並將其結果與Black Scholes之模型作一比較,此評價模型亦可視為再不完全市場下的另一選擇權評價模型.此外,我們亦觀察在漲跌幅限制下股價波動之行為,發現股價具有三點特徵,而Ornstein Uhlenbeck positon process比幾何布朗尼運動更能貼切的表現出這些特徵,因此採用Ornstein Uhlenbeck positon process之選擇權評價模型較能合適地評價在漲跌幅限制下之選擇權價值. / In this thesis, we extend the approach of Goldenberg (1986) to consider Ornstein-Uhlenbeck position process as an alternative to Geometric Brownian Motion in modeling the underlying asset prices, and construct the option pricing model with this process. Goldenberg (1986) argued that Ornstein-Uhlenbeck position process is more consistent with the observed future prices in imperfect markets, and it could express the correlation of stock prices. Our model is an alternative option pricing model in imperfect market. We also investigate the behavior of stock prices in markets with the imposition of price limits. We find that the use of Ornstein-Uhlenbeck position process is more consistent with the characteristics of stock prices with price limit constraints than Geometric Brownian Motion. The use of Ornstein-Uhlenbeck position process could provide a more concise closed form of option pricing model when considering price limit constraints.

<p>Brownian motion has met growing interest in mathematics, physics and particularly in finance since it was introduced in the beginning of the twentieth century. Stochastic processes generalizing Brownian motion have influenced many research fields theoretically and practically. Moreover, along with more refined techniques in measure theory and functional analysis more stochastic processes were constructed and studied. Lévy processes, with Brownian motionas a special case, have been of major interest in the recent decades. In addition, Lévy processes include a number of other important processes as special cases like Poisson processes and subordinators. They are also related to stable processes.</p><p>In this thesis we generalize a result by S. Chandrasekhar [2] and Edward Nelson who gave a detailed proof of this result in his book in 1967 [12]. In Nelson’s first result standard Ornstein-Uhlenbeck processes are studied. Physically this describes free particles performing a random and irregular movement in water caused by collisions with the water molecules. In a further step he introduces a nonlinear drift in the position variable, i.e. he studies the case when these particles are exposed to an external field of force in physical terms.</p><p>In this report, we aim to generalize the result of Edward Nelson to the case of α-stable Lévy processes. In other words we replace the driving noise of a standard Ornstein-Uhlenbeck process by an α-stable Lévy noise and introduce a scaling parameter uniformly in front of all vector fields in the cotangent space, even in front of the noise. This corresponds to time being sent to infinity. With Chandrasekhar’s and Nelson’s choice of the diffusion constant the stationary state of the velocity process (which is approached as time tends to infinity) is the Boltzmann distribution of statistical mechanics.The scaling limits we obtain in the absence and presence of a nonlinear drift term by using the scaling property of the characteristic functions and time change, can be extended to other types of processes rather than α-stable Lévy processes.</p><p>In future, we will consider to generalize this one dimensional result to Euclidean space of arbitrary finite dimension. A challenging task is to consider the geodesic flow on the cotangent bundle of a Riemannian manifold with scaled drift and scaled Lévy noise. Geometrically the Ornstein-Uhlenbeck process is defined on the tangent bundle of the real line and the driving Lévy noise is defined on the cotangent space.</p>