Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-06-14T13:27:03Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Hugo_Andrade_Doutorado.pdf: 3065927 bytes, checksum: 2eeb9c1ecb93e60c146992117b01cbb6 (MD5) / Made available in DSpace on 2016-06-14T13:27:03Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Hugo_Andrade_Doutorado.pdf: 3065927 bytes, checksum: 2eeb9c1ecb93e60c146992117b01cbb6 (MD5) Previous issue date: 2013-02-07 / CNPq / Nesta Dissertac¸ ˜ao estudamos a dinˆamica energ´etica das buscas aleat ´orias aplicadas ao problema de foraging, em que animais buscam por comida ou parceiros em ambientes escassos. Discutiremos, inicialmente, um modelo estat´ıstico de caminhadas aleat ´orias utilizando as distribuic¸ ˜oes de L´evy para os tamanhos dos passos de busca, as quais tˆem sido reportadas na literatura como estrat´egias de eficiˆencia ´otima para o problema. Em seguida vamos incluir no modelo ganhos e perdas de energia na caminhada aleat ´ oria de busca, e abordaremos a dinˆamica energ´etica do processo de busca unidimensional com extremos absorventes. Vamos discutir a transic¸ ˜ao de fase que o buscador experimenta de um estado ativo (“vivo”), t´ıpico de ambientes com abundˆancia de recursos, para um estado est´atico absorvente (“morto”), onde a busca ´e encerrada pela falta de energia oriunda do encontro de recursos. Obteremos os expoentes cr´ıticos relativos a essa transic¸ ˜ao atrav´es de abordagens te ´ oricas, tais como o m´etodo de primeira passagem para o estado de energia nula, e num´ericas, baseadas na hip´otese de escala. Mostraremos a independˆencia destes expoentes com a forma funcional da func¸ ˜ao gasto de energia. Por fim, faremos uma breve revis˜ao da literatura sobre a equac¸ ˜ao de Fokker-Planck canˆonica e tamb´em sobre as suas vers˜oes utilizando derivadas fracion´arias, numa prepararac¸ ˜ao para uma futura abordagem, durante o programa de Doutorado, do problema da busca aleat´oria envolvendo difus˜oes anˆomalas (por exemplo, superdifus˜ao) via equac¸ ˜oes diferenciais. / In this work we study the energy dynamics of random searches applied to the foraging problem, in which animals search for food or mates in scarce environments. Firstly, we discuss a statistical model of random search walks using the L´evy distribution of step lengths, which has been reported in the literature as an optimal solution to the problem. In the sequence we include in the model energy gains and losses during the search walk, and discuss the energy dynamics of the search process in a one dimensional space with absorbing boundaries. We discuss the phase transition that the searcher experiences from an active (“alive”) state, typical of environments abundant in resources, to a static absorbed (“dead”) one, in which the search is terminated due to the lack of energy obtained from the encounters.We obtain the critical exponents for this transition through both theoretical (such as the first-passage method to the state of zero energy) and numerical approaches, based on the scale hypothesis.We show the independence of the exponents with the functional form of the energy cost. Finally, we provide a brief review of the literature on the canonical Fokker-Planck equation and also on its version using fractional derivatives, in a preparation for a future approach of the random search problem involving anomalous diffusion (e.g., superdiffusion) through differential equations during the Ph.D. program.

This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.