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Tese (doutorado)—Universidade de Brasília, Instituto de Física, 2012. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2013-01-25T14:27:45Z No. of bitstreams: 1 2012_ReginaCeliaBuenoFonseca.pdf: 3069024 bytes, checksum: 9d0578691efea1b3ce2d08f432b428c2 (MD5) / Approved for entry into archive by Luanna Maia(luanna@bce.unb.br) on 2013-03-01T12:42:53Z (GMT) No. of bitstreams: 1 2012_ReginaCeliaBuenoFonseca.pdf: 3069024 bytes, checksum: 9d0578691efea1b3ce2d08f432b428c2 (MD5) / Made available in DSpace on 2013-03-01T12:42:53Z (GMT). No. of bitstreams: 1 2012_ReginaCeliaBuenoFonseca.pdf: 3069024 bytes, checksum: 9d0578691efea1b3ce2d08f432b428c2 (MD5) / Generalizamos o processo de Ornstein-Uhlenbeck (OU) usando o teorema de Doob. Relaxamos as condições de aussianidade e estacionariedade, assumindo um processo linear e homogêneo no tempo. A generalização proposta mantém muita da simplicidade do processo estocástico original, enquanto apresenta um comportamento mais rico. Os resultados analíticos são obtidos utilizando a pro- babilidade de transição e o formalismo da função característica e comparados com os dados empíricos do mercado de ações, que são notórios pelo comportamento não-estacionário e não-Gaussiano. As análises são focadas na forma do decaimento exponencial e na convergência assintótica dos quatro primeiros cumulantes considerando os retornos logarítmicos dos preços diários de ações. Mostramos que o modelo proposto oferece uma boa melhora em relação ao modelo OU clássico. ______________________________________________________________________________ ABSTRACT / We generalize the Ornstein-Uhlenbeck (OU) process using the Doob's theorem. We relax the Gaussian and stationary conditions, assuming a linear and time- homogeneous process. The proposed generalization retains much of the simplicity of the original stochastic process, while exhibiting a somewhat richer behavior. Analytical results are obtained using transition probability and the characteristic function formalism and compared with empirical stock market daily data, which are notorious for the non-stationary and non-Gaussian behavior. The analysis focus on the decay patterns and the convergence study of the rst four cumulants considering the logarithmic returns of stock prices. It is shown that the proposed model o ers a good improvement over the classical OU model.

Cette thèse est consacrée au problème d'estimation de la fonction de dérive de certains modèles de processus stochastiques périodiques lorsque la durée d'observation tend vers l'infini. Aucune hypothèse de récurrence n'est posée a priori.Dans un premier temps nous considérons le modèle du type signal plus bruit dζt = f (t, θ)dt + σ(t)dWt,; et puis nous étudions l'estimation du paramètre θ à partir d'une observation continue et puis d'une observation discrète du processus {ζt} sur l'intervalle [0; T]. Les fonctions f (·, ·) et σ(·) sont continues et périodiques en t de même période P > 0, σ(·) > 0 et θ ∈ Θ ⊂R. Nous établissons la convergence en probabilité d'un estimateur du maximum de vraisemblance θˆT , sa normalité asymptotique et son efficacité asymptotique minimax. Lorsque f (t, θ) = θf (t), l'expression de θˆT est explicite et nous obtenons la convergence en moyenne quadratique aussi bien pour le cas d'une observation continue que pour le cas d'une observation discrète. De plus, nous déduisons la convergence presque sûre dans le cas d'une observation continue.Dans la seconde partie nous traitons l'estimation non-paramétrique de la fonction f(_) pour les modèles périodiques du type signal plus bruit et du type Ornstein-Uhlenbeck donnés par dζt = f (t)dt + σ(t)dWt, dξt = f (t)ξtdt + dWt. Pour le premier modèle, un estimateur à noyau périodique est construit, la convergence en moyenne quadratique uniformément sur [0; P] et presque sûre de cet estimateur est établie ainsi que sa normalité asymptotique. Dans le cas du modèle d'Ornstein-Uhlenbeck, la convergence du biais ainsi que la convergence en moyenne quadratique uniformément sur [0; P] sont prouvées, et leurs vitesses de convergence sont étudiées. / In this thesis, we consider a drift estimation problem of a certain class of stochastic periodic processes when the length of observation goes to infinity. Firstly, we deal with the linear periodic signal plus noise model dζt = f (t, θ)dt + σ(t)dWt, ;and we study the parametric estimation from a continuous and discrete observation of the process f_tg throughout the interval [0; T]. Using the maximum likelihood method we show the existence of an estimator θˆT which is consistent, asymptotically normal and asymptotically efficient in the sens minimax. When f(t; _) = _f(t), the expression of ^_T is explicit and we obtain the mean square convergence in the both continuous and discrete observation cases. In addition, we deduce the strong consistency in the case of continuous observation.Secondly, we consider the nonparametric estimation problem of the function f(_) for the next two periodic models of type signal plus noise and Ornstein-Uhlenbeckd_t = f(t)dt + _(t)dWt; d_t = f(t)_tdt + dWt:For the signal plus noise model, we build a kernel estimator, the convergence in mean square uniformly over [0; P] and almost sure convergence are established, as well as the asymptotic normality. For the Ornstein-Uhlenbeck model, we prove the convergence uniformly over [0; P] of the bias and the mean square convergence. Moreover, we study the speed of these convergences.

Le projet s'inscrit dans la dynamique de systématisation statistique qui s'opère aujourd'hui dans le champ de l'écologie comparative. Les différents traits quantitatifs d'un jeu d'espèces échantillonné peuvent être vus comme le résultat d'un processus stochastique courant le long d'un arbre phylogénétique, ce qui permet de prendre en compte des corrélations issues d'histoires évolutives communes. Certains changements environnementaux peuvent produire un déplacement de niches évolutive, qui se traduisent par un saut dans la valeur du processus stochastique décrivant l'évolution au cours du temps du trait des espèces concernées. Parce qu'on ne mesure la valeur du processus dynamique qu'à un seul instant, pour les espèces actuelles, certains scénarii d'évolution ne peuvent être reconstruits, ou présentent des problèmes d'identifiabilité, que l'on étudie avec soin. On construit ici un modèle à données incomplètes d'inférence statistique, que l'on implémente efficacement. La position des sauts est détectée de manière automatique, et leur nombre est choisi grâce à une procédure de sélection de modèle adaptée à la structure du problème, et pour laquelle on dispose de certaines garanties théoriques. Un arbre phylogénétique ne prend pas en compte les phénomènes d'hybridation ou de transferts de gènes horizontaux, qui sont fréquents dans certains groupes d'organismes, comme les plantes ou les bactéries. Pour pallier ce problème, on utilise alors un réseau phylogénétique, pour lequel on propose une adaptation du modèle d'évolution de traits quantitatifs décrit précédemment. Ce modèle permet d'étudier l'hétérosis, qui se manifeste lorsqu'un hybride présente un trait d'une valeur exceptionnelle par rapport à celles de ses deux parents. / This project is aiming at taking a step further in the process of systematic statistical modeling that is occurring in the field of comparative ecology. A way to account for correlations between quantitative traits of a set of sampled species due to common evolutionary histories is to see the current state as the result of a stochastic process running on a phylogenetic tree. Due to environmental changes, some ecological niches can shift in time, inducing a shift in the parameters values of the stochastic process modeling trait evolution. Because we only measure the value of the process at a single time point, for extant species, some evolutionary scenarios cannot be reconstructed, or have some identifiability issues, that we carefully study. We construct an incomplete-data model for statistical inference, along with an efficient implementation. We perform an automatic shift detection, and choose the number of shifts thanks to a model selection procedure, specifically crafted to handle the special structure of the problem. Theoretical guaranties are derived in some special cases. A phylogenetic tree cannot take into account hybridization or horizontal gene transfer events, that are widely spread in some groups of species, such as plants or bacterial organisms. A phylogenetic network can be used to deal with these events. We develop a new model of trait evolution on this kind of structure, that takes non-linear effects such as heterosis into account. Heterosis, or hybrid vigor or depression, is a well studied effect, that happens when a hybrid species has a trait value that is outside of the range of its two parents.

The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very) well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a), which is applied by Perron (1989) to study the effects of an (assumed) exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992) to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003), which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996). An empirical application is provided. / Hoy en día es una práctica estándar de trabajo empírico la aplicación de diferentes estadísticos de contraste de raíz unitaria. A pesar de ser un aspecto práctico, estos estadísticos poseen distribuciones complejas y no estándar que dependen de funcionales de ciertos procesos estocásticos y sus derivaciones representan una barrera incluso para varios econometristas teóricos. Estas derivaciones están basadas en herramientas estadísticas fundamentales y rigurosas que no son (muy) bien conocidas por econometristas estándar. El presente artículo completa esta brecha al explicar en una forma simple una de estas herramientas fundamentales la cual es el Teorema del Límite Central Funcional. Por lo tanto, este documento analiza los fundamentos y la aplicabilidad de dos versiones del Teorema del Límite Central Funcional dentro del marco de una raíz unitaria con un quiebre estructural. La atención inicial se centra en la estructura probabilística de las series de tiempo propuesta por Phillips (1987a), la cual es aplicada por Perron (1989) para estudiar los efectos de un quiebre estructural (asumido) exógeno sobre la potencia de las pruebas Dickey-Fuller aumentadas y por Zivot y Andrews (1992) para criticar el supuesto de exogeneidad y proponer un método para estimar un punto de quiebre endógeno. Un método sistemático para tratar con aspectos de eficiencia es introducido por Perron y Rodríguez (2003), el cual extiende el enfoque de Mínimos Cuadrados Generalizados para eliminar los componentes determinísticos de Elliot et al. (1996). Se presenta además una aplicación empírica.

This dissertation studies a collection of problems on trading assets and derivatives over finite and infinite horizons. In the first part, we analyze an optimal switching problem with transaction costs that involves an infinite sequence of trades. The investor's value functions and optimal timing strategies are derived when prices are driven by an exponential Ornstein-Uhlenbeck (XOU) or Cox-Ingersoll-Ross (CIR) process. We compare the findings to the results from the associated optimal double stopping problems and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Our results show that when prices are driven by a CIR process, optimal strategies for the switching problems are of the classic buy-low-sell-high type. On the other hand, under XOU price dynamics, the investor should refrain from entering the market if the current price is very close to zero. As a result, the continuation (waiting) region for entry is disconnected. In both models, we provide numerical examples to illustrate the dependence of timing strategies on model parameters. In the second part, we study the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the OU, CIR or XOU model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing options to enter and exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short. Finally, we analyze the optimal risk-averse timing to sell a risky asset. The investor's risk preference is described by the exponential, power or log utility. Two stochastic models are considered for the asset price -- the geometric Brownian motion (GBM) and XOU models to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor's value function and certainty equivalent non-concave in price even though the utility function is concave in wealth. Numerical results are provided to illustrate the investor's optimal strategies and the premia associated with optimally timing to sell with different utilities under different price dynamics.

We obtain the characteristic relaxation rates and relaxation modes of a Bose gas in three regimes. The classical regime corresponds to a classical gas of hard spheres and the quantum regime corresponds to an interacting quantum Bose gas with no Bose-Einstein condensate present. In the condensed regime a Bose-Einstein condensate is present and modifies the behavior of the gas. In each regime there is a different kinetic equation that describes the evolution of the relevant distribution function. The classical kinetic equation is the Boltzmann equation and the quantum kinetic equation with no condensate present is the Uehling-Uhlenbeck equation. When a condensate is present, we derive a new kinetic equation that describes the evolution of the momentum distribution of Bogoliubov excitations or bogolons. For each of the three kinetic equations, we linearize the collision integral and use it to generate the elements of a collision matrix. The eigenvalues of this matrix give us the characteristic relaxation rates and the eigenvectors give us the relaxation modes. We report numerical results for the eigenvalues in each regime as the particle species, density and temperature of the gas are varied. / text

Ce mémoire porte sur la densité du premier temps de passage d'un processus gaussien et markovien à travers une frontière. Ce problème est résolu pour quelques cas particuliers, mais il n'est pas encore possible pour l'instant de le résoudre de façon analytique pour une frontière déterministe quelconque. (Di Nardo et al., 2001) ont proposé une méthode qui utilise des fonctions symétriques pour un ensemble de frontières qui généralisent celles de (Daniels, 1996). C'est ce qui est principalement étudié ici. De plus, deux exemples d'applications en finance sont considérés. Finalement, on regarde aussi un exemple de simulations pour comparer cette méthode à celle de (Durbin et Williams, 1992). ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Processus gaussien et markovien, mouvement brownien, processus d'Ornstein-Uhlenbeck, premier temps de passage.

Submitted by ARIEL TEIXEIRA (ateixeira@fgvmail.br) on 2014-03-14T15:38:35Z No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2014-03-14T16:29:51Z (GMT) No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Approved for entry into archive by Maria Almeida (maria.socorro@fgv.br) on 2014-04-09T14:58:17Z (GMT) No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Made available in DSpace on 2014-04-09T14:58:29Z (GMT). No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) Previous issue date: 2014-02-17 / Pair trading is an old and well-known technique among traders. In this paper, we discuss an important element not commonly debated in Brazil: the cointegration between pairs, which would guarantee the spread stability. We run the Dickey-Fuller test to check cointegration, and then compare the results with non-cointegrated pairs. We found that the Sharpe ratio of cointegrated pairs is greater than the non-cointegrated. We also use the Ornstein-Uhlenbeck equation in order to calculate the half-life of the pairs. Again, this improves their performance. Last, we use the leverage suggested by Kelly Formula, once again improving the results.

M.Sc.