Dans ce travail, nous étudions les propriétés statistiques de certains systèmes dynamiques déterministes et stochastiques. Nous nous intéressons particulièrement aux valeurs extrêmes et à la récurrence. Nous montrons l’existence de Lois pour les Valeurs Extrêmes(LVE) et pour les Statistiques des Temps d’Entrée (STE) et des Temps de Retour (STR) pour des systèmes avec décroissance des corrélations rapide. Nous étudions aussi la convergence du Processus Ponctuel d’Evènements Rares (PPER).Dans la première partie, nous nous intéressons aux systèmes dynamiques déterministes, et nous caractérisons complètement les propriétés précédentes dans le cas des systèmes dilatants. Nous montrons l’existence d’un Indice Extrême (IE) strictement plus petit que 1 autour des points périodiques, et qui vaut 1 dans le cas non-périodique, mettant ainsi en évidence une dichotomie dans la dynamique caractérisée par l’indice extrême. Dans un contexte plus général, nous montrons que le PPER converge soit vers une distribution de Poisson pour des points non-périodiques, soit vers une distribution de Poisson mélangée avec une distribution multiple de type géométrique pour des points périodiques. De plus, nous déterminons explicitement la limite des PPER autour des points de discontinuité et nous obtenons des distributions de Poisson mélangées avec des distributions multiples différentes de la distribution géométrique habituelle. Dans la deuxième partie, nous considérons des systèmes dynamiques stochastiques obtenus en perturbant de manière aléatoire un système déterministe donné. Nous élaborons deux méthodes nous permettant d’obtenir des lois pour les Valeurs Extrêmes et les statistiques de la récurrence en présence de bruits aléatoires. La première approche est de nature probabiliste tandis que la seconde nécessite des outils d’analyse spectrale. Indépendamment du point choisi, nous montrons que l’IE est constamment égal à 1 et que le PPER converge vers la distribution de Poisson standard. / In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson. / Neste trabalho, estudamos as propriedades estatısticas de sistemas dinâmicos deterministicos e estocasticos. Estamos particularmente interessados em valores extremos e recorrência. Provamos a existência de Leis de Valores Extremos (LVE) e Estatısticas doTempo de Entrada (ETE) / Estatısticas de Tempo de Retorno (ETR) para sistemas comdecaimento de correlaçoes contra observaveis em L1. Também realizamos o estudo daconvergência dos Processos Pontuais de Acontecimentos Raros (PPAR). Na primeira parte, investigamos o problema para dinâmica determinıstica e caracterizamos completamente o comportamento extremal de sistemas expansores. Mostramos que ha uma dicotomia quanto 00E0 existência de um Indice de Extrema (IE). Nomeadamente, provamos que o IE é estritamente menor do que 1 em torno de pontos periodicos e é igual a 1 para pontos aperiodicos. Num contexto mais geral, mostramos que os PPAR convergem para um processo de Poisson simples ou um processo de Poisson composto, em que a distribuiçao de multiplicidade é geométrica, dependendo se o centro é um ponto aperiodico ou periodico, respectivamente. Além disso, realizamos uma analise da convergência dos PPAR em pontos de descontinuidade, o que conduziu à descoberta de convergência para um processo de Poisson composto com uma distribuiçao de multiplicidade diferente da usual distribuiçao geométrica. Na segunda parte, consideramos dinâmica estocastica obtida por perturbaçao aleatoria de um sistema determinıstico por inclusao de um ruıdo aditivo. Apresentamos duas técnicas complementares que nos permitem obter LVE e as ETE na presen¸ca deste tipo de ruıdo. A primeira abordagem é mais probabilıstica enquanto que a outra usa sobretudo teoria espectral. Conclui-se que, independentemente do centro escolhido, o IE é sempre igual a 1 e os PPAR convergem para o processo de Poisson simples.

Considere um processo estocástico X_m em tempo discreto definido sobre o alfabeto finito A. Seja x_0^k-1 uma palavra fixa sobre A^k. No estudo das propriedades estatísticas na teoria de recorrência de Poincaré, é clássico o estudo do tempo decorrente até que a sequência fixa x_0^k-1 seja encontrada em uma realização do processo. Tipicamente, esta é uma quantidade exponencialmente grande com relação ao comprimento da palavra. Contrariamente, o primeiro tempo de retorno possível para uma sequência dada está definido como sendo o mínimo entre os tempos de entrada de todas as sequências que começam com a própria palavra e é uma quantidade tipicamente pequena, da ordem do tamanho da palavra. Neste trabalho estudamos o comportamento da probabilidade deste primeiro retorno possível de uma palavra x_0^k-1 dado que o processo começa com ela mesma. Esta quantidade mede a intensidade de que, uma vez observado um conjunto alvo, possam ser observados agrupamentos ou clusters. Provamos que, sob certas condições, a taxa de decaimento exponencial desta probabilidade converge para a entropia para quase toda a sequência quando k diverge. Apresentamos também um estimador desta probabilidade para árvores de contexto e mostramos sua consistência. / Considering a stochastic process X_m in a discrete defined time over a finite alphabet A and x_0^k-1 a fixed word over A^k. In the study of the statistical properties of the Poincaré recurrence theory, it is usual the study of the time elapsed until a fixed sequence x_0^k-1 appears in a given realization of process. This quantity is known as the hitting time and it is usually exponentially large in relation to the size of word. On the opposite, the first possible return time of a given word is defined as the minimum among all the hitting times of realizations that begins with the given word x_0^k-1. This quantity is tipically small that is of the order of the length of the sequence. In this work, we study the probability of the first possible return time given that the process begins of the target word. This quantity measures the intensity of that, once observed the target set, it can be observed in clusters. We show that, under certain conditions, the exponential decay rate of this probability converges to the entropy for all almost every word x_0^k-1 as k diverges. We also present an estimator of this probability for context trees and shows its consistency.

Considere um processo estocástico X_m em tempo discreto definido sobre o alfabeto finito A. Seja x_0^k-1 uma palavra fixa sobre A^k. No estudo das propriedades estatísticas na teoria de recorrência de Poincaré, é clássico o estudo do tempo decorrente até que a sequência fixa x_0^k-1 seja encontrada em uma realização do processo. Tipicamente, esta é uma quantidade exponencialmente grande com relação ao comprimento da palavra. Contrariamente, o primeiro tempo de retorno possível para uma sequência dada está definido como sendo o mínimo entre os tempos de entrada de todas as sequências que começam com a própria palavra e é uma quantidade tipicamente pequena, da ordem do tamanho da palavra. Neste trabalho estudamos o comportamento da probabilidade deste primeiro retorno possível de uma palavra x_0^k-1 dado que o processo começa com ela mesma. Esta quantidade mede a intensidade de que, uma vez observado um conjunto alvo, possam ser observados agrupamentos ou clusters. Provamos que, sob certas condições, a taxa de decaimento exponencial desta probabilidade converge para a entropia para quase toda a sequência quando k diverge. Apresentamos também um estimador desta probabilidade para árvores de contexto e mostramos sua consistência. / Considering a stochastic process X_m in a discrete defined time over a finite alphabet A and x_0^k-1 a fixed word over A^k. In the study of the statistical properties of the Poincaré recurrence theory, it is usual the study of the time elapsed until a fixed sequence x_0^k-1 appears in a given realization of process. This quantity is known as the hitting time and it is usually exponentially large in relation to the size of word. On the opposite, the first possible return time of a given word is defined as the minimum among all the hitting times of realizations that begins with the given word x_0^k-1. This quantity is tipically small that is of the order of the length of the sequence. In this work, we study the probability of the first possible return time given that the process begins of the target word. This quantity measures the intensity of that, once observed the target set, it can be observed in clusters. We show that, under certain conditions, the exponential decay rate of this probability converges to the entropy for all almost every word x_0^k-1 as k diverges. We also present an estimator of this probability for context trees and shows its consistency.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I: Martingale properties in time-homogeneous diffusion models: Part I of the thesis studies martingale properties of stock prices in stochastic volatility models driven by time-homogeneous diffusions. We find necessary and sufficient conditions for the martingale properties. The conditions are based on the local integrability of certain deterministic test functions. Part II: Analytical pricing methods in time-homogeneous diffusion models: Part II of the thesis studies probabilistic methods for determining the Laplace transform of the first hitting time of an integral functional of a time-homogeneous diffusion, and pricing an arithmetic Asian option when the stock price is modeled by a time-homogeneous diffusion. We also consider the pricing of discrete variance swaps and discrete gamma swaps in stochastic volatility models based on time-homogeneous diffusions. Part III: Nearly Unbiased Monte Carlo Simulation: Part III of the thesis studies the unbiased Monte Carlo simulation of option prices when the characteristic function of the stock price is known but its density function is unknown or complicated.

A credit derivative is a financial instrument whose value depends on the credit risk of an underlying asset or assets. Credit risk is the possibility that the obligor fails to honor any payment obligation. This thesis proposes four new computational methods for the valuation of credit derivatives. Compared with synthetic collateralized debt obligations (CDOs) or basket default swaps (BDS), the value of which depends on the defaults of a prescribed underlying portfolio, a forward-starting CDO or BDS has a random underlying portfolio, as some ``names'' may default before the CDO or BDS starts. We develop an approach to convert a forward product to an equivalent standard one. Therefore, we avoid having to consider the default combinations in the period between the start of the forward contract and the start of the associated CDO or BDS. In addition, we propose a hybrid method combining Monte Carlo simulation with an analytical method to obtain an effective method for pricing forward-starting BDS. Current factor copula models are static and fail to calibrate consistently against market quotes. To overcome this deficiency, we develop a novel chaining technique to build a multi-period factor copula model from several one-period factor copula models. This allows the default correlations to be time-dependent, thereby allowing the model to fit market quotes consistently. Previously developed multi-period factor copula models require multi-dimensional integration, usually computed by Monte Carlo simulation, which makes the calibration extremely time consuming. Our chaining method, on the other hand, possesses the Markov property. This allows us to compute the portfolio loss distribution of a completely homogeneous pool analytically. The multi-period factor copula is a discrete-time dynamic model. As a first step towards developing a continuous-time dynamic model, we model the default of an underlying by the first hitting time of a Wiener process, which starts from a random initial state. We find an explicit relation between the default distribution and the initial state distribution of the Wiener process. Furthermore, conditions on the existence of such a relation are discussed. This approach allows us to match market quotes consistently.

The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I: Martingale properties in time-homogeneous diffusion models: Part I of the thesis studies martingale properties of stock prices in stochastic volatility models driven by time-homogeneous diffusions. We find necessary and sufficient conditions for the martingale properties. The conditions are based on the local integrability of certain deterministic test functions. Part II: Analytical pricing methods in time-homogeneous diffusion models: Part II of the thesis studies probabilistic methods for determining the Laplace transform of the first hitting time of an integral functional of a time-homogeneous diffusion, and pricing an arithmetic Asian option when the stock price is modeled by a time-homogeneous diffusion. We also consider the pricing of discrete variance swaps and discrete gamma swaps in stochastic volatility models based on time-homogeneous diffusions. Part III: Nearly Unbiased Monte Carlo Simulation: Part III of the thesis studies the unbiased Monte Carlo simulation of option prices when the characteristic function of the stock price is known but its density function is unknown or complicated.

A credit derivative is a financial instrument whose value depends on the credit risk of an underlying asset or assets. Credit risk is the possibility that the obligor fails to honor any payment obligation. This thesis proposes four new computational methods for the valuation of credit derivatives. Compared with synthetic collateralized debt obligations (CDOs) or basket default swaps (BDS), the value of which depends on the defaults of a prescribed underlying portfolio, a forward-starting CDO or BDS has a random underlying portfolio, as some ``names'' may default before the CDO or BDS starts. We develop an approach to convert a forward product to an equivalent standard one. Therefore, we avoid having to consider the default combinations in the period between the start of the forward contract and the start of the associated CDO or BDS. In addition, we propose a hybrid method combining Monte Carlo simulation with an analytical method to obtain an effective method for pricing forward-starting BDS. Current factor copula models are static and fail to calibrate consistently against market quotes. To overcome this deficiency, we develop a novel chaining technique to build a multi-period factor copula model from several one-period factor copula models. This allows the default correlations to be time-dependent, thereby allowing the model to fit market quotes consistently. Previously developed multi-period factor copula models require multi-dimensional integration, usually computed by Monte Carlo simulation, which makes the calibration extremely time consuming. Our chaining method, on the other hand, possesses the Markov property. This allows us to compute the portfolio loss distribution of a completely homogeneous pool analytically. The multi-period factor copula is a discrete-time dynamic model. As a first step towards developing a continuous-time dynamic model, we model the default of an underlying by the first hitting time of a Wiener process, which starts from a random initial state. We find an explicit relation between the default distribution and the initial state distribution of the Wiener process. Furthermore, conditions on the existence of such a relation are discussed. This approach allows us to match market quotes consistently.

In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson.