Small-time moment asymptotics for Lévy processes

Figueroa-López, José E.

doi:10.1016/j.spl.2008.07.012

Cited by 49 publications

(43 citation statements)

References 12 publications

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“…A proof of (1.3) for a general Lévy process can be found in [29] (see his Corollary 8.9). Let us remark that (1.3) is also valid for certain unbounded functions ϕ, which does not necessarily vanish in a neighborhood of the origin, but rather converge to 0 at a proper rate (see [15] for more details). The second key property is related to the decomposition of X into two independent processes: one accounting for the "small" jumps and a compound Poisson process collecting the "big" jumps.…”

Section: S(x)dx ∀A ∈ B(r\{0})mentioning

confidence: 99%

“…To achieve this goal, we need to impose some regularity on either the Lévy process or the moment functions ϕ. Following the second approach, [15] shed light on this problem for functions ϕ ∈ C 2 b (R); namely, twice-continuously differentiable functions ϕ such that lim sup |x|→∞ |ϕ…”

Section: Proposition 21 Let ϕ Be ν-Continuous Bounded and Such Thatmentioning

confidence: 99%

“…The following result can be found in [15] (see Proposition 3.1), where a proof is provided for a certain class of unbounded functions ϕ:…”

Section: Proposition 21 Let ϕ Be ν-Continuous Bounded and Such Thatmentioning

confidence: 99%

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Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

Figueroa-López¹

2009

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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A Lévy 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 infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling 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 piece-wise 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 necessary to attain the optimal minimax rate is explicitly identified. The proposed method is illustrated by simulation experiments in the case of variance Gamma processes.

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Section: S(x)dx ∀A ∈ B(r\{0})mentioning

confidence: 99%

Section: Proposition 21 Let ϕ Be ν-Continuous Bounded and Such Thatmentioning

confidence: 99%

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Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

Figueroa-López¹

2009

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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“…The Lévy measure of (L t ) has a density given by Proof. For the first point, we refer to [30]. For the second point, the assumptions and the fact that r ≤ 1 imply…”

Section: Let L T = B γ T Where (γ T ) Is a Pure Jump Increasing Lévy mentioning

confidence: 99%

“…In the latter case, authors distinguish between low frequency data (sampling interval ∆ is fixed) or high frequency data (∆ tends to 0). We concentrate in this chapter on high frequency data setting since it is simpler and allows to consider several adaptive estimation methods: deconvolution with cut-off selection, contrast penalization, see our works [17], [19], [20], and also [30], [31], [62] and adaptive kernels (see Section 4.3, and also [7]). …”

Section: Bibliographic Commentsmentioning

confidence: 99%

Adaptive Estimation for Lévy Processes

Comte

Genon-Catalot

2014

Lévy Matters IV

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Parametric Estimation of Lévy Processes

Masuda

2014

Lecture Notes in Mathematics

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The main purpose of this chapter is to present some theoretical aspects of parametric estimation of Lévy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models. Asymptotics for several classes of explicit estimating functions are discussed. In addition to the asymptotic normality at several rates of convergence, a uniform tail-probability estimate for statistical random fields is given. As specific cases, we discuss method of moments for the stable Lévy processes in much greater detail, with briefly mentioning locally stable Lévy processes too. Also discussed is, due to its theoretical importance, a brief review of how the classical likelihood approach works or does not, beyond the fact that the likelihood function is not explicit.

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Small-time moment asymptotics for Lévy processes

Cited by 49 publications

References 12 publications

Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

Adaptive Estimation for Lévy Processes

Parametric Estimation of Lévy Processes

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