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.

Wassily Hoeffding, in his 1963 paper, introduces a procedure to derive inequalities between distributions. This method relies on finding a martingale coupling between the two random variables. I have developed a construction that establishes such couplings in various urn models. I use this construction to prove the inequality between the hypergeometric and binomial random variables that appears in Hoeffding's paper. I have then used and extended my urn construction to create new inequalities.

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.

Cette thèse est consacrée à l'étude de certains modèles de déformation semi-paramétriques. Notre objectif est de proposer des méthodes récursives, issues d'algorithmes stochastiques, pour estimer les paramètres de ces modèles. Dans la première partie, on présente les outils théoriques existants qui nous seront utiles dans la deuxième partie. Dans un premier temps, on présente un panorama général sur les méthodes d'approximation stochastique, en se focalisant en particulier sur les algorithmes de Robbins-Monro et de Kiefer-Wolfowitz. Dans un second temps, on présente les méthodes à noyaux pour l'estimation de fonction de densité ou de régression. On s'intéresse plus particulièrement aux deux estimateurs à noyaux les plus courants qui sont l'estimateur de Parzen-Rosenblatt et l'estimateur de Nadaraya-Watson, en présentant les versions récursives de ces deux estimateurs.Dans la seconde partie, on présente tout d'abord une procédure d'estimation récursive semi-paramétrique du paramètre de translation et de la fonction de régression pour le modèle de translation dans la situation où la fonction de lien est périodique. On généralise ensuite ces techniques au modèle vectoriel de déformation à forme commune en estimant les paramètres de moyenne, de translation et d'échelle, ainsi que la fonction de régression. On s'intéresse finalement au modèle de déformation paramétrique de variables aléatoires dans le cadre où la déformation est connue à un paramètre réel près. Pour ces trois modèles, on établit la convergence presque sûre ainsi que la normalité asymptotique des estimateurs paramétriques et non paramétriques proposés. Enfin, on illustre numériquement le comportement de nos estimateurs sur des données simulées et des données réelles.

Thèse doctorat : Mathématiques : La Rochelle : 2003. / Bibliogr. p. 113-119.

Thesis (Ph.D.)--University of Wollongong, 2008. / Typescript. Includes bibliographical references: leaf 239-254.

American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.

Orientador: Serguei Popov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:42:00Z (GMT). No. of bitstreams: 1 Costa_MarceloRocha_M.pdf: 576974 bytes, checksum: 3b01a9f15e6e0f9fdd98631dc69cd202 (MD5) Previous issue date: 2014 / Resumo: O problema considerado no presente trabalho cumpre o papel de reforçar a eficácia dos métodos apresentados nos capítulos introdutórios, bem como investiga a resposta de um problema até então não publicado na literatura especializada. Introduzimos uma partícula realizando um passeio aleatório simples no espaço, ou seja, uma partícula que a cada passo escolhe uniformemente um de seus vizinhos para onde irá saltar. Fixando sua posição inicial ao longo da fronteira do cubo, pergunta-se: qual é a probabilidade de que a trajetória de tal partícula nunca mais retorne ao cubo? Em outras palavras, se T é o tempo de primeiro retorno ao cubo, estamos interessados em descrever o comportamento assintótico da probabilidade de que T seja infinito / Abstract: It has been considered in this work a problem which play a role of showing the effectiveness of the content covered in the introductory chapters, as well as it is a unsolved problem across the specialized literature. We introduce a particle performing a simple random walk in space, i.e., a particle which at each step choose uniformly one of its neighbourhood sites to which it then jumps into. Fixed its initial position along the boundary of a cube, we are interested in answering the following question: what is the probability that such particle's trajectory will never reach the cube again. In other words, if T is the first return time to the cube, we aim to analyse the asymptotic behaviour of the probability that T is infinite / Mestrado / Estatistica / Mestre em Estatística

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T03:35:11Z (GMT). No. of bitstreams: 1 Silva_FabianoBorgesda_M.pdf: 388532 bytes, checksum: 847fc3b7dce8c11700ac92aff1ce3c34 (MD5) Previous issue date: 2005 / Resumo: Este trabalho tem por finalidade explorar resultados de aplicacoes harmonicas, atraves do calculo estocastico em variedades. Esta organizado da seguinte forma: Nos dois primeiros capitulos sao introduzidos conceitos e resultados sobre calculo estocastico no Rn, geometria diferencial e grupos de Lie. No terceiro capitulo temos as definicoes de aplicacoes harmonicas e a equacao de Euler-Lagrange. E finalmente, no ultimo, damos uma caracterizacao para aplicacoes harmonicas atraves de martingales, que sera importante para explorar alguns resultados sobre aplicacoes harmonicas do ponto de vista do calculo estocastico em variedades / Abstract: In this work we explore results of harmonic mappings, via stochastic calculus in manifolds. The text is organized as follows: In the first two chapters, we introduce concepts and results about stochastic calculus in Rn, differential geometry and Lie groups. In the third chapter we have the definitions of harmonic mappings and the Euler-Lagrange equation. Finally, in the last chapter, we give a characterization of harmonic mappings via martingales, this will be important to explore some results about harmonic mappings from the point of view of stochastic calculus in manifolds / Mestrado / Matematica / Mestre em Matemática

La première partie de cette thèse porte sur l’étude de la marche aléatoire sur le tore Td := Rd/Zd définie par une mesure de probabilité SLd(Z). Pour étudier le Théorème Central Limite et la loi du logarithme itéré, nous appliquons la méthode de Gordin qui consiste à se ramener à des martingales. Pour cela, nous utilisons un résultat de Bourgain, Furmann, Lindenstrauss et Mozes nous permettant de résoudre l’équation de Poisson pour des points ayant de bonnes propriétés diophantiennes. Dans la deuxième partie, nous étudions la marche sur Rd\{0} définie par l’action de SLd(R) et nous montrons un résultat de vitesse de convergence dans le théorème de renouvellement de Guivarc’h et Le Page. / The first part of this thesis deals with the random walk on the torus Td := Rd/Zd defined by a robability measure on SLd(Z). To study the Central Limit Theorem and the Law of the Iterated Logarithm, we apply Gordin’s method. To do so, we use a result proved by Bourgain, Furmann, Lindenstrauss and Mozes to solve Poisson’s equation at point’s having good diophantine properties.In the second part, we study the walk on Rd \ {0} defined by the action of SLd(R) and we prove a result about the rate of convergence in Guivarc’h and Le Page’s renewal theorem.