# Schémas de discrétisation anticipatifs et estimation du paramètre de dérive d'une diffusion

• Volume: 4, page 233-258
• ISSN: 1292-8100

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## Abstract

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Let YT = (Yt)t∈[0,T] be a real ergodic diffusion process which drift depends on an unkown parameter ${\theta }_{0}\in {ℝ}^{p}$. Our aim is to estimate θ0 from a discrete observation of the process YT, (Ykδ)k=0,n, for a fixed and small δ, as T = nδ goes to infinity. For that purpose, we adapt the Generalized Method of Moments (see Hansen) to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of θ0 with a bias of order δ2 (resp. δ4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's [1], which were obtained for a Gaussian diffusion process, which drift is linear in θ.

## How to cite

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Samos, Sandie Souchet. "Schémas de discrétisation anticipatifs et estimation du paramètre de dérive d'une diffusion." ESAIM: Probability and Statistics 4 (2010): 233-258. <http://eudml.org/doc/197771>.

@article{Samos2010,
abstract = { Let YT = (Yt)t∈[0,T] be a real ergodic diffusion process which drift depends on an unkown parameter $\theta_\{0\}\in \mathbb\{R\}^\{p\}$. Our aim is to estimate θ0 from a discrete observation of the process YT, (Ykδ)k=0,n, for a fixed and small δ, as T = nδ goes to infinity. For that purpose, we adapt the Generalized Method of Moments (see Hansen) to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of θ0 with a bias of order δ2 (resp. δ4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's [1], which were obtained for a Gaussian diffusion process, which drift is linear in θ. },
author = {Samos, Sandie Souchet},
journal = {ESAIM: Probability and Statistics},
keywords = {Schéma du trapèze; schéma de Simpson; schéma anticipatif; diffusion ergodique; estimation par variables instrumentales; méthode des moments généralisés; contraste; biais d'estimation; efficacité asymptotique en variance.; trapezoidal scheme; Simpson scheme; ergodic diffusion; instrumental variables estimation; generalized method of moments; contrast; bias of estimation; variance asymptotic efficiency},
language = {eng},
month = {3},
pages = {233-258},
publisher = {EDP Sciences},
title = {Schémas de discrétisation anticipatifs et estimation du paramètre de dérive d'une diffusion},
url = {http://eudml.org/doc/197771},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Samos, Sandie Souchet
TI - Schémas de discrétisation anticipatifs et estimation du paramètre de dérive d'une diffusion
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 233
EP - 258
AB - Let YT = (Yt)t∈[0,T] be a real ergodic diffusion process which drift depends on an unkown parameter $\theta_{0}\in \mathbb{R}^{p}$. Our aim is to estimate θ0 from a discrete observation of the process YT, (Ykδ)k=0,n, for a fixed and small δ, as T = nδ goes to infinity. For that purpose, we adapt the Generalized Method of Moments (see Hansen) to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of θ0 with a bias of order δ2 (resp. δ4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's [1], which were obtained for a Gaussian diffusion process, which drift is linear in θ.
LA - eng
KW - Schéma du trapèze; schéma de Simpson; schéma anticipatif; diffusion ergodique; estimation par variables instrumentales; méthode des moments généralisés; contraste; biais d'estimation; efficacité asymptotique en variance.; trapezoidal scheme; Simpson scheme; ergodic diffusion; instrumental variables estimation; generalized method of moments; contrast; bias of estimation; variance asymptotic efficiency
UR - http://eudml.org/doc/197771
ER -

## References

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