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1 
Rare events and conditional limit theorems for a class of spectrally positive, heavytailed Lévy processes /Richardson, Gregory Scott, January 2000 (has links)
Thesis (Ph. D.)University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 6770). Available also in a digital version from Dissertation Abstracts.

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Study of Gaussian processes, Lévy processes and infinitely divisible distributionsVeillette, Mark S. January 2011 (has links)
Thesis (Ph.D.)Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at openhelp@bu.edu. Thank you. / In this thesis, we study distribution functions and distributionalrelated quantities for various stochastic processes and probability distributions, including Gaussian processes, inverse Levy subordinators, Poisson stochastic integrals, nonnegative infinitely divisible distributions and the Rosenblatt distribution. We obtain analytical results for each case, and in instances where no closed form exists for the distribution, we provide numerical solutions. We mainly use two methods to analyze such distributions. In some cases, we characterize distribution functions by viewing them as solutions to differential equations. These are used to obtain moments and distributions functions of the underlying random variables. In other cases, we obtain results using inversion of Laplace or Fourier transforms. These methods include the PostWidder inversion formula for Laplace transforms, and Edgeworth approximations. In Chapter 1, we consider differential equations related to Gaussian processes. It is well known that the heat equation together with appropriate initial conditions characterize the marginal distribution of Brownian motion. We generalize this connection to finite dimensional distributions of arbitrary Gaussian processes. In Chapter 2, we study the inverses of Levy subordinators. These processes are nonMarkovian and their finitedimensional distributions are not known in closed form. We derive a differential equation related to these processes and use it to find an expression for joint moments. We compute numerically these joint moments in Chapter 3 and include several examples. Chapter 4 considers Poisson stochastic integrals. We show that the distribution function of these random variables satisfies a KolmogorovFeller equation, and we describe the regularity of solutions and numerically solve this equation. Chapter 5 presents a technique for computing the density function or distribution function of any nonnegative infinitely divisible distribution based on the PostWidder method. In Chapter 6, we consider a distribution given by an infinite sum of weighted gamma distributions. We derive the LevyKhintchine representation and show when the tail of this sum is asymptotically normal. We derive a BerryEssen bound and Edgeworth expansions for its distribution function. Finally, in Chapter 7 we look at the Rosenblatt distribution, which can be expressed as a infinite sum of weighted chisquared distributions. We apply the expansions in Chapter 6 to compute its distribution function. / 20310101

3 
Lévy processes in inverse problemsFlenner, Arjuna, January 2004 (has links)
Thesis (Ph. D.)University of MissouriColumbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 105115). Also available on the Internet.

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Lévy processes in inverse problems /Flenner, Arjuna, January 2004 (has links)
Thesis (Ph. D.)University of MissouriColumbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 105115). Also available on the Internet.

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PARAMETER ESTIMATION FOR GEOMETRIC L EVY PROCESSES WITH STOCHASTIC VOLATILITYUnknown Date (has links)
In finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alphastable Levy motion to the process related to dynamics of logreturns in the BlackScholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter estimation for discretely observed stochastic differential equations driven by Levy noises. We also discuss the consistency and asymptotic properties of the proposed estimators. Simulation results of the model are also presented to show the validity of the estimators. We then propose a new model where the volatility is not a constant. We consider generalized alphastable geometric Levy processes where the stochastic volatility follows the CoxIngersollRoss (CIR) model in Cox et al. [9]. A number of methods have been proposed for estimating parameters for stable laws. However, a complication arises in estimation of the parameters in our model because of the presence of the unobservable stochastic volatility. To combat this complication we use the sample characteristic function method proposed by Press [48] and the conditional least squares method as mentioned in Overbeck and Ryden [47] to estimate all the parameters. We then discuss the consistency and asymptotic properties of the proposed estimators and establish a Central Limit Theorem. We perform simulations to assess the validity of the estimators. We also present several tables to show the comparison of estimators using different choices of arguments ui's. We conclude that all the estimators converge as expected regardless of the choice of ui's. / Includes bibliography. / Dissertation (Ph.D.)Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

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Stochastic volatility modelsLe, Truc January 2005 (has links)
Abstract not available

7 
Nonparametric estimation of Lévy processes with a view towards mathematical financeFigueroaLópez, José Enrique. January 2003 (has links) (PDF)
Thesis (Ph. D.)Mathematics, Georgia Institute of Technology, 2004. / Marcus C. Spruill, Committee Member ; Richard Serfozo, Committee Member ; Shijie Deng, Committee Member ; Christian Houdre, Committee Chair ; Robert P. Kertz, Committee Member. Vita. Includes bibliographical references.

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Pricing of European options using empirical characteristic functionsBinkowski, Karol Patryk. January 2008 (has links)
Thesis (PhD)Macquarie University, Division of Economic and Financial Studies, Dept. of Statistics, 2008. / Bibliography: p. 7377.

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Pricing of European options using empirical characteristic functionsBinkowski, Karol Patryk January 2008 (has links)
Thesis (PhD)Macquarie University, Division of Economic and Financial Studies, Dept. of Statistics, 2008. / Bibliography: p. 7377. / Introduction  Lévy processes used in option pricing  Option pricing for Lévy processes  Option pricing based on empirical characteristic functions  Performance of the five models on historical data  Conclusions  References  Appendix A. Proofs  Appendix B. Supplements  Appendix C. Matlab programs. / Pricing problems of financial derivatives are among the most important ones in Quantitative Finance. Since 1973 when a Nobel prize winning model was introduced by Black, Merton and Scholes the Brownian Motion (BM) process gained huge attention of professionals professionals. It is now known, however, that stock market logreturns do not follow the very popular BM process. Derivative pricing models which are based on more general Lévy processes tend to perform better. Carr & Madan (1999) and Lewis (2001) (CML) developed a method for vanilla options valuation based on a characteristic function of asset logreturns assuming that they follow a Lévy process. Assuming that at least part of the problem is in adequate modeling of the distribution of logreturns of the underlying price process, we use instead a nonparametric approach in the CML formula and replaced the unknown characteristic function with its empirical version, the Empirical Characteristic Functions (ECF). We consider four modifications of this model based on the ECF. The first modification requires only historical logreturns of the underlying price process. The other three modifications of the model need, in addition, a calibration based on historical option prices. We compare their performance based on the historical data of the DAX index and on ODAX options written on the index between the 1st of June 2006 and the 17th of May 2007. The resulting pricing errors show that one of our models performs, at least in the cases considered in the project, better than the Carr & Madan (1999) model based on calibration of a parametric Lévy model, called a VG model. Our study seems to confirm a necessity of using implied parameters, apart from an adequate modeling of the probability distribution of the asset logreturns. It indicates that to precisely reproduce behaviour of the real option prices yet other factors like stochastic volatility need to be included in the option pricing model. Fortunately the discrepancies between our model and real option prices are reduced by introducing the implied parameters which seem to be easily modeled and forecasted using a mixture of regression and time series models. Such approach is computationaly less expensive than the explicit modeling of the stochastic volatility like in the Heston (1993) model and its modifications. / Mode of access: World Wide Web. / x, 111 p. ill., charts

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