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SELF-SIMILARITY AND LAMPERTI CONVERGENCE FOR FAMILIES OF STOCHASTIC PROCESSES

JORGENSEN, Bent; MARTINEZ, Jose R.; DEMETRIO, Clarice G. B.
Fonte: SPRINGER Publicador: SPRINGER
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
56.31%
We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to certain important families of processes that are not self-similar in the conventional sense. This includes Hougaard Levy processes such as the Poisson processes, Brownian motions with drift and the inverse Gaussian processes, and some new fractional Hougaard motions defined as moving averages of Hougaard Levy process. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.; Danish Natural Science Research Council; FAPESP, Brazil

Modelagem de séries temporais financeiras multidimensionais via processos estocásticos e cópulas de Lévy ; Multidimensional Financial Time Series Modelling via Lévy Stochastic Processes and Copulas

Santos, Edson Bastos e
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 16/12/2005 PT
Relevância na Pesquisa
56.53%
O principal objetivo deste estudo é descrever modelos para séries temporais de ativos financeiros que sejam robustos às tradicionais hipóteses: distribuição gaussiana e continuidade. O primeiro capítulo está preocupado em apresentar, de uma maneira geral, os conceitos matemáticos mais importantes relacionadas a processos estocásticos e difusões. O segundo capítulo trata de processos de incrementos independentes e estacionários, i.e., processos de Lévy, suas trajetórias estocásticas, propriedades distribucionais e, a relação entre processos markovianos e martingales. Alguns dos resultados apresentados neste capítulo são: a estrutura e as propriedades dos processos compostos de Poisson, medida de Lévy, decomposição de Lévy-Itô e representação de Lévy-Khinchin. O terceiro capítulo mostra como construir processos de Lévy por meio de transformações lineares, inclinação da medida de Lévy e subordina ção. Uma atenção especial é dada aos processos subordinados, tais como os modelos variância gama, normal gaussiana invertida e hiperbólico generalizado. Neste capítulo também é apresentado um exemplo pragmático com dados brasileiros de estimação de parâmetros por meio do método de máxima Verossimilhança. O quarto capítulo é devotado aos modelos multidimensionais e...

Ensaios em finanças quantitativas: apreçamento de derivativos multidimensionais via processos de Lévy, e topologia e propagação do risco sistêmico; Essays in quantitative finance: multidimensional derivative pricing via Lévy processes, and systemic risk topology na risk propagation

Santos, Edson Bastos e
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 24/03/2010 PT
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66.43%
Este estudo contempla dois ensaios em finanças quantitativas, relacionados, respectivamente, a modelos de apreçamento e risco sistêmico. No Capitulo 1, e apresentado uma alternativa para modelar opções multidimensionais, cujas estruturas de ganhos e perdas dependam das trajetórias dos processos dos preços dos ativos objetos. A modelagem sugerida considera os processos de Levy, uma classe de processos estocásticos bastante ampla, que permite a existência de saltos (descontinuidades) no processo dos preços dos ativos financeiros, e tem como caso particular o movimento Browniano. Para escrever a dependência entre os processos, os conceitos estáticos de copulas ordinárias são estendidos para o contexto dos processos de Levy, levando em consideração a medida de Levy, que caracteriza o comportamento dos saltos. São realizados estudos comparativos entre as copulas dinâmicas de Clayton e de Frank, no apreçamento dos contratos derivativos do tipo asiático, utilizando-se processos gama e técnicas de simulação de Monte Carlo. No Capitulo 2, a estrutura e dinâmica interbancária das exposições mutuas entre as instituições financeiras no Brasil e explorada bem como o capital destas reservas, utilizando um conjunto de dados únicos que considera vários períodos entre 2007 e 2008. Para isto e mostrado que a rede de exposições pode ser modelada adequadamente como um gráfico estocástico dirigido de escala - livre (ponderada) seguindo distribuições que apresentam caudas grossas. A relação entre as conexões das instituições financeiras e seu colchão-de-capital também são investigados neste estudo. Finalmente...

Duality with time-changed Lévy processes

Fajardo, José
Fonte: Escola de Pós-Graduação em Economia da FGV Publicador: Escola de Pós-Graduação em Economia da FGV
Tipo: Relatório
EN_US
Relevância na Pesquisa
66.18%
In this paper we study the pricing problem of derivatives written in terms of a two dimensional time{changed L¶evy processes. Then, we examine an existing relation between prices of put and call options, of both the European and the American type. This relation is called put{call duality. It includes as a particular case, the relation known as put{call symmetry. Necessary and su±cient conditions for put{call symmetry to hold are shown, in terms of the triplet of local charac- teristic of the Time{changed L¶evy process. In this way we extend the results obtained in Fajardo and Mordecki (2004) to the case of time{changed Lévy processes.

Stochastic volatility jump-diffusion models as time-changed Lévy processes

Matos, Ricardo Nuno Santos Aleixo de
Fonte: Universidade de Lisboa Publicador: Universidade de Lisboa
Tipo: Dissertação de Mestrado
Publicado em //2014 ENG
Relevância na Pesquisa
56.33%
Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2014; Esta tese foca-se na aplicação da técnica de time-changed Lévy processes, apresentada em primeiro lugar por Carr and Wu (2004), a fim de deduzir o modelo de Bakshi et al. (1997) com uma distribuição arbitrária do tamanho do salto. O segundo objectivo passa por obter um modelo com correlação total, depois de deduzir o teorema fundamental onde se obtém a função característica conjunta de um número finito de time-changed Lévy processes sob a medida de alavancagem neutra. Posteriormente, obtivémos a função característica exacta para o preço de um activo com volatilidade estocástica, taxas de juros estocásticas, saltos e correlação total. Tanto quanto sabemos, foi a primeira vez que se obteve a função característica exacta de um modelo com volatilidade estocástica, taxas de juros estocásticas, saltos e correlação total.; This thesis focuses on applying the time-changed Lévy processes technique firstly presented by Carr and Wu (2004) in order to deduce the Bakshi et al. (1997) model with a general jump size distribution. The second goal is reach a full correlation scheme, after reaching the fundamental theorem...

Integro-differential equations for option pricing in exponential Lévy models

Cruz, José Manuel Teixeira Santos
Fonte: Instituto Superior de Economia e Gestão Publicador: Instituto Superior de Economia e Gestão
Tipo: Dissertação de Mestrado
Publicado em //2013 ENG
Relevância na Pesquisa
56.3%
Mestrado em Matemática Financeira; This dissertation discusses under which conditions we can express the function that represents the option price as the solution of a certain partial integro-differential equation (PIDE) in a exponential Lévy model. The main difference between this case and the Black Scholes case is that there is a non-local term in the equation, which makes the analysis more complicated. Also, we discuss under which conditions we can obtain a Feynman-Kac formula for the case of a pure jump process and discuss the conditions under which option prices are classical solutions of the PIDEs. When such conditions are not verified, we consider the concept of viscosity solutions which only requires that the function representing the option price is continuous. Continuity results for option prices of barrier options are presented for some types of Lévy processes. In addition, we show the same continuity results for processes of finite variation and with no diffusion component. Also, we present some examples in which the function that represents the option price is discontinuous. Moreover, we present a numerical scheme that gives the price of an European put option for the Variance Gamma process. This finite difference scheme was initially proposed by Cont and Voltchkova...

Avaliação de opções com processos de Lévy e transformações temporais

Martins, Nuno Filipe Costa
Fonte: Instituto Superior de Economia e Gestão Publicador: Instituto Superior de Economia e Gestão
Tipo: Dissertação de Mestrado
Publicado em //2014 POR
Relevância na Pesquisa
56.25%
Mestrado em Matemática Financeira; O objetivo do presente trabalho é responder à questão de investigação: como avaliar opções exóticas através de processos de Lévy com transformações temporais? Para o efeito, analisa-se o modelo de referência CGMY-Gamma-OU (processo de Lévy com transformação temporal) e compara-se a performance deste modelo, ao nível da calibração a dados reais de mercado e avaliação de opções exóticas, com os modelos (sem transformação temporal) de Black-Scholes e CGMY. A opção exótica em análise consiste numa opção pouco explorada na literatura financeira, e com importantes implicações teóricas ao nível das estratégias de cobertura de risco (hedging), designada de opção sobre os momentos de ordem k de um ativo financeiro subjacente. Demonstra-se que, dependendo da aproximação dos modelos aos dados de mercado e do momento de ordem k, o modelo CGMY-Gamma-OU constitui uma boa alternativa na avaliação da opção exótica referida.; The objective of this study is to answer the research question: how to evaluate exotic options through time-changed Lévy processes? For this purpose, we consider the reference model CGMY-Gamma-OU (time-changed Lévy process) and a comparison is made between the performance of this model...

Optimal consumption and investment with Lévy processes

Barbachan,José Fajardo
Fonte: Fundação Getúlio Vargas Publicador: Fundação Getúlio Vargas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/12/2003 EN
Relevância na Pesquisa
56.1%
We study the intertemporal consumption and investment problem in a continuous time setting when the security prices follow a Geometric Lévy process. Using stochastic calculus for semimartingales we obtain conditions for the existence of optimal consumption policies. Also, we give a charaterization of the equivalent martingale measures.

Equivalent martingale measures and Lévy processes

Fajardo,José Santiago
Fonte: Fundação Getúlio Vargas Publicador: Fundação Getúlio Vargas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/12/2006 EN
Relevância na Pesquisa
56.1%
In this paper we compute equivalent martingale measures when the asset price returns are modelled by a Lévy process. We follow the approach introduced by Gerber and Shiu (1994).

Feller Processes: The Next Generation in Modeling. Brownian Motion, Lévy Processes and Beyond

Böttcher, Björn
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Publicado em 03/12/2010 EN
Relevância na Pesquisa
46.36%
We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Lévy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Lévy processes, of which Brownian Motion is a special case, have become increasingly popular. Lévy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Lévy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.

Adaptive Lévy Processes and Area-Restricted Search in Human Foraging

Hills, Thomas T.; Kalff, Christopher; Wiener, Jan M.
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Publicado em 05/04/2013 EN
Relevância na Pesquisa
46.38%
A considerable amount of research has claimed that animals’ foraging behaviors display movement lengths with power-law distributed tails, characteristic of Lévy flights and Lévy walks. Though these claims have recently come into question, the proposal that many animals forage using Lévy processes nonetheless remains. A Lévy process does not consider when or where resources are encountered, and samples movement lengths independently of past experience. However, Lévy processes too have come into question based on the observation that in patchy resource environments resource-sensitive foraging strategies, like area-restricted search, perform better than Lévy flights yet can still generate heavy-tailed distributions of movement lengths. To investigate these questions further, we tracked humans as they searched for hidden resources in an open-field virtual environment, with either patchy or dispersed resource distributions. Supporting previous research, for both conditions logarithmic binning methods were consistent with Lévy flights and rank-frequency methods–comparing alternative distributions using maximum likelihood methods–showed the strongest support for bounded power-law distributions (truncated Lévy flights). However...

Small-time Chung Laws for L evy processes

Phelan, Thomas Michael
Fonte: Universidade Nacional da Austrália Publicador: Universidade Nacional da Austrália
Tipo: Thesis (MPhil); Master of Philosophy (MPhil)
EN_AU
Relevância na Pesquisa
56.09%
In this thesis we review and add to the literature extending the so-called `other' law of the iterated logarithm of Chung (1948). By adapting the large-time techniques of Rushton (2007) to the small-time setting and employing and slightly extending a characterisation result of Maller and Mason (2008), we derive both one-dimensional and functional Chung laws for a large class of Levy processes lying in the domain of attraction of strictly stable laws at zero. In particular, our results extend the work of Buchmann and Maller (2011) to encompass processes with vanishing Gaussian component lying in the domain of attraction of a normal distribution at zero.; Supervisor: Boris Buchmann; Yes

Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics

Ben Salah, Zied
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation
EN
Relevância na Pesquisa
46.44%
Cette thèse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance. Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problème de ruine et de notions fondamentales des mathématiques financières. Le deuxième chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problème de ruine pour un processus de risque markovien additif. Une identification de systèmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type". Le troisième chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article présente une extension de l'espérance de la fonction de pénalité actualisée pour un processus subordinateur de risque perturbé par un mouvement brownien. Cette extension contient une série de fonctions escomptée éspérée des minima successives dus aux sauts du processus de risque après la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisée pour déterminer la valeur espérée du capital d'injection actualisé. Finallement...

An almost sure functional limit theorem at zero for a class of Levy processes normed by the square root function, and applications

Buchmann, Boris; Maller, Ross; Szimayer, Alex
Fonte: Springer Publicador: Springer
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.15%
A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X(t) for which lim supt↓0 |X (t)|/√t ∈ (0, ∞) occurs almost surely (a.s.). For such processes we have a kind o

Meromorphic Levy processes and their fluctuation identities

Kuznetsov, Alexey; Kyprianou, Andreas E.; Pardo, Juan Carlos
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative Levy processes, hyper-exponential and generalized hyper-exponential Levy processes, Lamperti-stable processes, Hypergeometric processes, Beta-processes and Theta-processes. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Levy-Khintchin exponent, all of which appear on the imaginary axis of the complex plane. The specific structure of the M-class Wiener-Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance...

Integral Equations in the Theory of Levy Processes

Sakhnovich, Lev
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/02/2007
Relevância na Pesquisa
46.37%
In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussian process, the Meixner process, the compound Poisson process).

Applying the Wiener-Hopf Monte Carlo simulation technique for Levy processes to path functionals such as first passage times, undershoots and overshoots

Ferreiro-Castilla, Albert; van Schaik, Kees
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.37%
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Levy model) and insurance (ruin time, debt at ruin and related quantities for a Levy insurance risk process). The technique works for any Levy process whose running infimum and supremum evaluated at an independent exponential time allows sampling from. This includes classic examples such as stable processes, subclasses of spectrally one sided Levy processes and large new families such as meromorphic Levy processes. Finally we present some examples. A particular aspect that is illustrated is that the WHMC simulation technique performs much better at approximating first passage times than a `plain' Monte Carlo simulation technique based on sampling increments of the Levy process.

The dichotomy of recurrence and transience of semi-Levy processes

Maejima, Makoto; Takamune, Taisuke; Ueda, Yohei
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/09/2012
Relevância na Pesquisa
46.38%
Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the laws of large numbers for semi-Levy processes.

Levy processes with summable Levy measures, long time behavior

Sakhnovich, Lev
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/03/2014
Relevância na Pesquisa
46.41%
In our previous paper (ArXiv:1306.1492) we have proved that a representation of the infinitesimal generators $L$ for Levy processes $X_t$ can be written down in a convolution type form. For the case of non-summable Levy measures we constructed the quasi-potential operators $B$ and investigated the long time behavior of $X_t$. In the present paper we consider Levy processes $X_t$ with summable Levy measures. In this case the form of the quasi-potential operators $B$ essentially differs from the form in the case of non-summable Levy measures. We use this new form in order to study the long time behavior of $X_t$ for the case of summable Levy measures.; Comment: This paper is a further development of our previous paper (ArXiv:1306.1492), where Levy processes with non-summable Levy measures were studied. Here we study Levy processes with summable Levy measures

Moments of passage times for Levy processes

Doney, R A; Maller, Ross
Fonte: Gauthier-Villars Publicador: Gauthier-Villars
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.21%
We give necessary and sufficient conditions, in terms of characteristics of the process, for finiteness of moments of passage times of general Lévy processes above horizontal, linear or certain curved boundaries. They apply in particular to processes whi