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54
Approximation of Small Jumps of L¶evy Processes with a View Toward Simulation
 Journal of Applied Probability
, 2001
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Hyperbolic Processes in Finance
, 2001
"... Distributions that have tails heavier than the normal distribution are ubiquitous in finance. For purposes such as risk management and derivative pricing it is important to use relatively simple models that can capture the heavy tails and other relevant features of financial data. A class of distrib ..."
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Cited by 24 (6 self)
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Distributions that have tails heavier than the normal distribution are ubiquitous in finance. For purposes such as risk management and derivative pricing it is important to use relatively simple models that can capture the heavy tails and other relevant features of financial data. A class of distributions that is very often able to fit
Nonparametric estimation for Levy processes with a view towards mathematical finance, 2004, available at ArXiv math.ST/0412351 33 Finkelestein M., Tucker H.G. and Veeh J.A., Extinguishing the distinguished logarithm problems
, 1997
"... Nonparametric methods for the estimation of the Lévy density of a Lévy process X are developed. Estimators that can be written in terms of the “jumps ” of X are introduced, and so are discretedata based approximations. A model selection approach made up of two steps is investigated. The first step c ..."
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Cited by 16 (10 self)
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Nonparametric methods for the estimation of the Lévy density of a Lévy process X are developed. Estimators that can be written in terms of the “jumps ” of X are introduced, and so are discretedata based approximations. A model selection approach made up of two steps is investigated. The first step consists in the selection of a good estimator from a linear model of proposed Lévy densities, while the second is a datadriven selection of a linear model among a given collection of linear models. By providing lower bounds for the minimax risk of estimation over Besov Lévy densities, our estimators are shown to achieve the “best ” rate of convergence. A numerical study for the case of histogram estimators and for variance Gamma processes, models of key importance in risky asset price modeling driven by Lévy processes, is presented. 1
Simulation methods for Levydriven CARMA stochastic volatility models
 Journal of Business and Economic Statistics
, 2006
"... We develop simulation schemes for the new classes of nonGaussian pure jump L¶evy processes for stochastic volatility. We write the price and volatility processes as integrals against a vector L¶evy process, which then makes series approximation methods directly applicable. These methods entail simu ..."
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Cited by 15 (0 self)
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We develop simulation schemes for the new classes of nonGaussian pure jump L¶evy processes for stochastic volatility. We write the price and volatility processes as integrals against a vector L¶evy process, which then makes series approximation methods directly applicable. These methods entail simulation of the L¶evy increments and formation of weighted sums of the increments; they do not require a closedform expression for a tail mass function nor speci¯cation of a copula function. We also present a new, and apparently quite °exible, bivariate mixture of gammas model for the driving L¶evy process. Within this setup, it is quite straightforward to generate simulations from a L¶evydriven CARMA stochastic volatility model augmented by a purejump price component. Simulations reveal the wide range of di®erent types of ¯nancial price processes that can be generated in this manner, including processes with persistent stochastic volatility, dynamic leverage, and jumps.
Nonparametric estimation for Lévy models based on discretesampling
 IMS Lecture Notes  Monograph Series. Optimality: The Third Erich L. Lehmann Symposium
"... Abstract: ALévy model combines a Brownian motion with drift and a purejump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinitedimensional parameter controlling the jump dynamics of the process, is studied under a discretesampling scheme. In that ..."
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Cited by 14 (7 self)
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Abstract: ALévy model combines a Brownian motion with drift and a purejump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinitedimensional parameter controlling the jump dynamics of the process, is studied under a discretesampling scheme. In that case, the jumps are latent variables whose statistical properties can in principle be assessed when the frequency of observations increase to infinity. We propose nonparametric estimators for the Lévy density following Grenander’s method of sieves. The associated problem of selecting a suitable approximating sieve is subsequently investigated using regular piecewise polynomials as sieves and assuming standard smoothness conditions on the Lévy density. By sampling the process at a high enough frequency relative to the time horizon T, we show that it is feasible to choose the dimension of the sieve so that the rate of convergence of the risk of estimation off the origin is the best possible from a minimax point of view, and even if the estimation were based on the whole sample path of the process. The sampling frequency
The approximate Euler method for Lévy driven stochastic differential equations
, 2005
"... 1) Approximating Markov process expectations. In applications of Markov processes, it is frequently necessary to compute IE(g(Xt)), where X is the process modelling the system of interest. While this expectation can sometimes be obtained by direct numerical computation, for example, by applying nume ..."
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Cited by 13 (1 self)
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1) Approximating Markov process expectations. In applications of Markov processes, it is frequently necessary to compute IE(g(Xt)), where X is the process modelling the system of interest. While this expectation can sometimes be obtained by direct numerical computation, for example, by applying numerical schemes for partial differential
Gaussian approximation of multivariate Levy processes with applications to simulation of tempered and operator stable processes
, 2005
"... Problem of simulation of multivariate Levy processes is investigated. The method based on shot noise series expansions of such processes combined with Gaussian approximation of the remainder is established in full generality. Formulas that can be used for simulation of tempered stable, operator stab ..."
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Cited by 10 (1 self)
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Problem of simulation of multivariate Levy processes is investigated. The method based on shot noise series expansions of such processes combined with Gaussian approximation of the remainder is established in full generality. Formulas that can be used for simulation of tempered stable, operator stable and other multivariate processes are obtained.
CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators
, 2006
"... We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the onesided stable (Y/2) subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable (1/2) subordinator. The r ..."
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Cited by 8 (0 self)
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We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the onesided stable (Y/2) subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable (1/2) subordinator. The required time changes may be generated by simulating the requisite onesided stable subordinator and throwing away some of the jumps as described in Rosinski (2001). 1
Hölder regularity for operator scaling stable random fields
, 2007
"... Abstract. We investigate the sample paths regularity of operator scaling αstable random fields. Such fields were introduced in [6] as anisotropic generalizations of selfsimilar fields and satisfy the scaling property {X(c E x);x ∈ R d} (fdd) = {c H X(x);x ∈ R d} where E is a d ×d real matrix and H ..."
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Cited by 7 (3 self)
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Abstract. We investigate the sample paths regularity of operator scaling αstable random fields. Such fields were introduced in [6] as anisotropic generalizations of selfsimilar fields and satisfy the scaling property {X(c E x);x ∈ R d} (fdd) = {c H X(x);x ∈ R d} where E is a d ×d real matrix and H> 0. In the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of R d with respect to E. In particular, the directional Hölder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling random αstable random fields, with α ∈ (0,2) and d ≥ 2, the sample paths are almost surely discontinous. 1.