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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
Português
Relevância na Pesquisa
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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

Bivariate gamma-geometric law and its induced Levy process

Barreto-Souza, Wagner
Fonte: ELSEVIER INC; SAN DIEGO Publicador: ELSEVIER INC; SAN DIEGO
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
67.49432%
In this article we introduce a three-parameter extension of the bivariate exponential-geometric (BEG) law (Kozubowski and Panorska, 2005) [4]. We refer to this new distribution as the bivariate gamma-geometric (BGG) law. A bivariate random vector (X, N) follows the BGG law if N has geometric distribution and X may be represented (in law) as a sum of N independent and identically distributed gamma variables, where these variables are independent of N. Statistical properties such as moment generation and characteristic functions, moments and a variance-covariance matrix are provided. The marginal and conditional laws are also studied. We show that BBG distribution is infinitely divisible, just as the BEG model is. Further, we provide alternative representations for the BGG distribution and show that it enjoys a geometric stability property. Maximum likelihood estimation and inference are discussed and a reparametrization is proposed in order to obtain orthogonality of the parameters. We present an application to a real data set where our model provides a better fit than the BEG model. Our bivariate distribution induces a bivariate Levy process with correlated gamma and negative binomial processes, which extends the bivariate Levy motion proposed by Kozubowski et al. (2008) [6]. The marginals of our Levy motion are a mixture of gamma and negative binomial processes and we named it BMixGNB motion. Basic properties such as stochastic self-similarity and the covariance matrix of the process are presented. The bivariate distribution at fixed time of our BMixGNB process is also studied and some results are derived...

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 Português
Relevância na Pesquisa
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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...

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 Português
Relevância na Pesquisa
47.2385%
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...

On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for Lévy processes

Omidi Firouzi, Hassan
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation
Português
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Dans cette thèse, nous étudions quelques problèmes fondamentaux en mathématiques financières et actuarielles, ainsi que leurs applications. Cette thèse est constituée de trois contributions portant principalement sur la théorie de la mesure de risques, le problème de l’allocation du capital et la théorie des fluctuations. Dans le chapitre 2, nous construisons de nouvelles mesures de risque cohérentes et étudions l’allocation de capital dans le cadre de la théorie des risques collectifs. Pour ce faire, nous introduisons la famille des "mesures de risque entropique cumulatifs" (Cumulative Entropic Risk Measures). Le chapitre 3 étudie le problème du portefeuille optimal pour le Entropic Value at Risk dans le cas où les rendements sont modélisés par un processus de diffusion à sauts (Jump-Diffusion). Dans le chapitre 4, nous généralisons la notion de "statistiques naturelles de risque" (natural risk statistics) au cadre multivarié. Cette extension non-triviale produit des mesures de risque multivariées construites à partir des données financiéres et de données d’assurance. Le chapitre 5 introduit les concepts de "drawdown" et de la "vitesse d’épuisement" (speed of depletion) dans la théorie de la ruine. Nous étudions ces concepts pour des modeles de risque décrits par une famille de processus de Lévy spectrallement négatifs.; The aim of this thesis is to study fundamental problems in financial and insurance mathematics particularly the problem of measuring risk and its application within financial and insurance frameworks. The main contributions of this thesis can be classified in three main axes: the theory of risk measures...

Expansion of Lévy process functionals and its application in econometric estimation.

Dong, Chaohua
Fonte: Universidade de Adelaide Publicador: Universidade de Adelaide
Tipo: Tese de Doutorado
Publicado em //2012 Português
Relevância na Pesquisa
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This research focuses on the estimation of a class of econometric models for involved unknown nonlinear functionals of nonstationary processes. The proxy of nonstationary processes studied here is Lévy processes including Brownian motion as a particular one. A Lévy process is a càdlàg stochastic process which starts at zero almost surely, which has independent increments over disjoint intervals, which has stationary increment distribution meaning that under shift the distributions of increments are identical, which has stochastic continuous trajectory. Obviously, Brownian motion, Poisson process, Gamma process and Pascal process are fundamental examples of Lévy processes. Lévy processes (Z(t); t >0) studied in this thesis possess density or probability distribution functions which verify some properties stated in the text.; Thesis (Ph.D.) -- University of Adelaide, School of Economics, 2012

Distinguished limits of Lévy-Stable processes, and applications to option pricing

Cartea, Álvaro; Howison, Sam
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: info:eu-repo/semantics/submittedVersion; info:eu-repo/semantics/workingPaper Formato: application/pdf
Publicado em 19/08/2003 Português
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In this paper we derive analytic expressions for the value of European Put and Call options when the stock process follows an exponential Lévy-Stable process. It is shown that the generalised Black-Scholes operator for the Lévy-Stable case can be obtained as an asymptotic approximation of a process where the random variable follows a Damped- Lévy process. Finally, it is also shown that option prices under the Lévy-Stable case generate the volatility smile encountered in the financial markets when the Black-Scholes framework is employed; The first author acknowledges financial support from JP Morgan

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance

Cartea, Álvaro; Howison, Sam
Fonte: Taylor & Francis Publicador: Taylor & Francis
Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article Formato: application/pdf
Publicado em /06/2009 Português
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We show how to calculate European-style option prices when the log-stock price process follows a Lévy-Stable process with index parameter 1≤α≤2 and skewness parameter -1≤β≤1. Key to our result is to model integrated variance as an increasing Lévy-Stable process with continuous paths in Τ

Avaliação de derivados de catástrofe

Dinis, Marta Filipa Gomes
Fonte: Instituto Superior de Economia e Gestão. Publicador: Instituto Superior de Economia e Gestão.
Tipo: Dissertação de Mestrado
Publicado em /10/2011 Português
Relevância na Pesquisa
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Mestrado em Ciências Actuariais; Nas últimas décadas observou-se o desenvolvimento de produtos financeiros destinados a cobrir riscos catastróficos. Os derivados de catástrofe são instrumentos financeiros baseados num índice de perdas, como por exemplo o índice PCS, que reflecte as perdas da indústria seguradora ocorridas num determinado período. A presente dissertação pretende estudar duas metodologias de avaliação de derivados de catástrofe utilizando técnicas da transformada de Fourier para determinar uma fórmula analítica para o preço das opções transaccionadas. O índice de perdas na primeira metodologia (modelo de Mürmann) é modelado como um processo de Poisson composto, qualquer que seja o instante em análise até à maturidade. No segundo modelo de avaliação (modelo de Biagini, Bregman e Meyer-Brandis) é feita a distinção entre o período de perdas, durante o qual o evento catastrófico pode ocorrer, e o período de desenvolvimento, no qual as perdas registadas no período anterior são reestimadas. Supõe-se que o índice de perdas é modelado por um processo de Poisson composto no período de perdas e as perdas são reestimadas por um factor obtido por um processo de Lévy exponencial. Os modelos propostos são aplicados a dados de opções call spread...

A Multinomial Approximation for American Option Prices in Levy Process Models

Maller, Ross; Solomon, David H; Szimayer, Alexander
Fonte: Blackwell Publishing Ltd Publicador: Blackwell Publishing Ltd
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A Multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial

Finite approximation schemes for Levy processes, and their application to optimal stopping problems

Szimayer, Alexander; Maller, Ross
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
Português
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This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0, T], and having a finite number of states, for a pure jump Lévy process Lt. The sequences of discrete processes converge to the original process,

Stochastic integration for fractional Levy process and stochastic differential equation driven by fractional Levy noise

Lu, Xuebin; Dai, Wanyang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/07/2013 Português
Relevância na Pesquisa
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In this paper, based on the white noise analysis of square integrable pure-jump Levy process given by [1], we define the formal derivative of fractional Levy process defined by the square integrable pure-jump Levy process as the fractional Levy noises by considering fractional Levy process as the generalized functional of Levy process, and then we define the Skorohod integral with respect to the fractional Levy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional Levy noises and investigate the existence and uniqueness of their solutions; In addition, we propose a class of stochastic differential equations driven by fractional Levy noises and prove that under the Lipschtz and linear conditions there exists unique stochastic distribution-valued solution.; Comment: 16 pages, Accepted by ACTA MATHEMATICA SCIENTIA (A): Chinese Series (Chinese Journal of Mathematical Physics), 2013

Levy process simulation by stochastic step functions

Sørensen, Torquil Macdonald; Benth, Fred Espen
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/10/2011 Português
Relevância na Pesquisa
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We study a Monte Carlo algorithm for simulation of probability distributions based on stochastic step functions, and compare to the traditional Metropolis/Hastings method. Unlike the latter, the step function algorithm can produce an uncorrelated Markov chain. We apply this method to the simulation of Levy processes, for which simulation of uncorrelated jumps are essential. We perform numerical tests consisting of simulation from probability distributions, as well as simulation of Levy process paths. The Levy processes include a jump-diffusion with a Gaussian Levy measure, as well as jump-diffusion approximations of the infinite activity NIG and CGMY processes. To increase efficiency of the step function method, and to decrease correlations in the Metropolis/Hastings method, we introduce adaptive hybrid algorithms which employ uncorrelated draws from an adaptive discrete distribution defined on a space of subdivisions of the Levy measure space. The nonzero correlations in Metropolis/Hastings simulations result in heavy tails for the Levy process distribution at any fixed time. This problem is eliminated in the step function approach. In each case of the Gaussian, NIG and CGMY processes, we compare the distribution at t=1 with exact results and note the superiority of the step function approach.; Comment: 20 pages...

On exact sampling of the first passage event of Levy process with infinite Levy measure and bounded variation

Chi, Zhiyi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/07/2012 Português
Relevância na Pesquisa
47.45685%
We present an exact sampling method for the first passage event of a Levy process. The idea is to embed the process into another one whose first passage event can be sampled exactly, and then recover the part belonging to the former from the latter. The method is based on several distributional properties that appear to be new. We obtain general procedures to sample the first passage event of a subordinator across a regular non-increasing boundary, and that of a process with infinite Levy measure, bounded variation, and suitable drift across a constant level or interval. We give examples of application to a rather wide variety of Levy measures.

Cramer's estimate for a reflected Levy process

Doney, R. A.; Maller, R. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/05/2005 Português
Relevância na Pesquisa
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The natural analogue for a Levy process of Cramer's estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Levy process with finite negative mean which satisfies Cramer's condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of ``high excursions.''; Comment: Published at http://dx.doi.org/10.1214/105051605000000016 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

On the properites of Poisson random measures associated with a G-Levy process

Paczka, Krzysztof
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/11/2014 Português
Relevância na Pesquisa
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In this paper we study the properties of the Poisson random measure and the Poisson integral associated with a G-Levy process. We prove that a Poisson integral is a G-Levy process and give the conditions which ensure that a Poisson integral belongs to a good space of random variables. In particular, we study the relation between the quasi- continuity of an integrand and the quasi-continuity of the integral. Lastly, we apply the results to establish the pathwise decomposition of a G-Levy process into a generalized G-Brownian motion and a pure-jump G-Levy process and prove that both processes belong to a good space of random variables.

Financial Modeling and Option Theory with the Truncated Levy Process

Matacz, Andrew
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/10/1997 Português
Relevância na Pesquisa
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In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider, for the early Levy dominated regime, the issue of option hedging for two different hedging strategies that are in some sense optimal. These are compared with the usual delta hedging approach and found to differ significantly. I also derive the natural generalization of the Black-Scholes option pricing formula when the underlying security is modeled by a geometric TLP. This generalization would not be possible without the truncation.; Comment: 21 pages in Latex, 6 eps figures

A continous-time GARCH process driven by a Levy process: stationarity and second-order behaviour

Kluppelberg, Claudia; Lindner, Alexander; Maller, Ross
Fonte: Applied Probability Trust Publicador: Applied Probability Trust
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
67.264575%
We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1, 1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our 'COGARCH' (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

Cramers estimate for a reflected Levy process

Doney, R A; Maller, Ross
Fonte: Institute of Mathematical Statistics Publicador: Institute of Mathematical Statistics
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
57.022993%
The natural analogue for a Lévy process of Cramér's estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this esti

The Process of Creation of a Work of Literature and its Reception - The Creation of a Translation / O Processo de Criação de uma Obra Literária e sua Recepção - A Criação de uma Tradução

Levý, Jiří; Masaryk’s University; Althoff, Gustavo; Universidade Federal de Santa Catarina; Vidal, Cristiane; Universidade Federal de Santa Catarina
Fonte: Universidade Federal de Santa Catarina Publicador: Universidade Federal de Santa Catarina
Tipo: info:eu-repo/semantics/article; info:eu-repo/semantics/publishedVersion; Formato: application/pdf
Publicado em 31/07/2012 Português
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http://dx.doi.org/10.5007/1980-4237.2012n11p97Esse texto corresponde à seção I de três de uma versão condensada de um capítulo (de livro) intitulado “Gen­eze a recepce literárního díla” (The Process of Creation of a Work of Literature and its Reception) [O Processo de Criação de uma Obra Literária e sua Recepção], escrito por Levý em 1967 um pouco antes de sua morte e publicado numa coletânea de textos seus intitulada “Bude literární věda exaktní vědou?” (Will Literary Studies become an exact Discipline?) [Os Estudos Literários tornar-se-ão uma disciplina exata?], em 1971. Trata-se de uma reformulação de seu célebre artigo “Translation as a decision process” [A Tradução como um Processo de Decisão] publicado em inglês, também em 1967, alguns meses antes. Levý importou desse texto boa parte dos exemplos e do conteúdo que aparecem nessa seção I – The Creation of a Translation [A Criação de uma Tradução] –, havendo feito modificações e melhorias quanto à apresentação e argumentação. O leitor familiarizado com seu texto anterior poderá constatar o quanto tal estudo, um tanto esquelético e especulativo, foi aprimorado em tão curto espaço de tempo. Tal como naquele, nesse texto Levý conceitua e revela como o processo de criação de uma tradução literária se dá com base nas decisões do tradutor. Apresenta-se aqui sua tradução indireta ao português a partir do inglês.