Estimating functions for SDE driven by stable Lévy processes

Clément, Emmanuelle; Gloter, Arnaud

doi:10.1214/18-aihp920

Cited by 11 publications

(14 citation statements)

References 22 publications

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“…In the case α ≤ 1, our bound is of the same order of magnitude as the one presented by Clément and Gloter (2018) and Amorino and Gloter (2019). Furthermore, our result may also be applied in the case ρ > α − 1.…”

Section: Preliminary Resultssupporting

confidence: 71%

Rate-optimal estimation of the Blumenthal-Getoor index of a Lévy process

Mies¹

2019

Preprint

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The Blumenthal-Getoor (BG) index characterizes the jump measure of an infinitely active Lévy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations only achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.

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Section: Preliminary Resultssupporting

confidence: 71%

Rate-optimal estimation of the Blumenthal-Getoor index of a Lévy process

Mies¹

2019

Preprint

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“…The second and the third point of the lemma here above are proved in Lemma 4.5 of [9], while the first point is proved in Theorem 4.1 [9] and it shows us, using the exponential formula for Poisson measure, that h n is the function that turns our measure µ n into the measure associated to an α-stable process truncated with the function τ . Thus (L α,n t ) t∈[0,1] is a Lévy process with jump intensity ω → τ (ω∆ 1 α n ) |ω| 1+α and we recognize the law of an α-stable truncated process.…”

Section: The Function Hmentioning

confidence: 87%

“…In this section, we recall some results on Malliavin calculus for jump processes. We refer to [8] for a complete presentation and to [9] for the adaptation to our framework. We will work on the Poisson space associated to the measure µ n defining the process (L n t ) t∈[0,1] of the previous section, assuming that n is fixed.…”

Section: Malliavin Calculusmentioning

confidence: 99%

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

Amorino

Gloter

2020

Stochastic Processes and their Applications

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The problem of integrated volatility estimation for the solution X of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations in both short and long time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results the condition β > 1 2(2−α) on β (that is such that ( 1 n ) β is the threshold of the truncated quadratic variation) and on the degree of jump activity α was needed to have the original truncated realized volatility well-performed (see [22], [13]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple (α, β).Lévy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.

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“…Other relevant work includes [28] who propose theorems and estimating function methods for stable Lévy-driven sdes; [29], who prove error bounds between Lévy-driven sdes with small jumps and their Gaussian approximations; [30] provide Bayesian inference procedures using a quasi-likelihood approach and MCMC; [31], [32] provide a Multi-level Monte Carlo approach for evaluation of expectations, using coupled Euler approximations; and [33] who propose Euler schemes under Poisson-arrival times. In contrast to these approaches, we tackle the problem without resort to Euler approximations or pseudo-likelihood functions, and our methods are applicable to low-or high-frequency observations, but are currently limited to linear sdes.…”

Section: Summary Of Methods and Contributionmentioning

confidence: 99%

The Lévy State Space Model

Godsill

Riabiz

Kontoyiannis

2019

2019 53rd Asilomar Conference on Signals, Systems, and Computers

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In this paper we introduce a new class of state space models based on shot-noise simulation representations of non-Gaussian Lévy-driven linear systems, represented as stochastic differential equations. In particular a conditionally Gaussian version of the models is proposed that is able to capture heavytailed non-Gaussianity while retaining tractability for inference procedures. We focus on a canonical class of such processes, the α-stable Lévy processes, which retain important properties such as self-similarity and heavy-tails, while emphasizing that broader classes of non-Gaussian Lévy processes may be handled by similar methodology. An important feature is that we are able to marginalise both the skewness and the scale parameters of these challenging models from posterior probability distributions. The models are posed in continuous time and so are able to deal with irregular data arrival times. Example modelling and inference procedures are provided using Rao-Blackwellised sequential Monte Carlo applied to a two-dimensional Langevin model, and this is tested on real exchange rate data.

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Estimating functions for SDE driven by stable Lévy processes

Cited by 11 publications

References 22 publications

Rate-optimal estimation of the Blumenthal-Getoor index of a Lévy process

Rate-optimal estimation of the Blumenthal-Getoor index of a Lévy process

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

The Lévy State Space Model

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