Results 1  10
of
44
Approximation of small jumps of Lévy processes with a view towards simulation
 Journal of Applied Probability
, 2001
"... ..."
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 ..."
Abstract

Cited by 15 (10 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
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.
Sievebased confidence intervals and bands for Lévy densities
 Bernoulli
"... Abstract: A Lévy process combines a Brownian motion 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 considered here under a discretesampling scheme. In th ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract: A Lévy process combines a Brownian motion 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 considered here under a discretesampling scheme. In that case, the jumps are latent variables which statistical properties can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the Lévy density based on Grenander’s method of sieves had been proposed in [11]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the Lévy density. In the pointwise case, we find feasible estimators which converge to s at a rate that is arbitrarily close to the rate of the minimax risk of estimation for smooth Lévy densities. We determine how frequently one needs to sample to attain the desired rate. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to log −1/2 (n) · n −1/3, where n is the number of observations. The rate is valid provided that s is smooth enough, and that the time horizon Tn and the dimension of the sieve are appropriately chosen in terms of n.
On the Lamperti stable processes
, 802
"... We consider a new family of IRdvalued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance [6], [10], [23] and [28]). This family of processes shares many properties with the temper ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We consider a new family of IRdvalued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance [6], [10], [23] and [28]). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosiński [31] and Houdré and Kawai [16] respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.. Key words and phrases: Lamperti stable distributions and processes, stable