Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations

Renaud Marty

ESAIM: Probability and Statistics (2010)

  • Volume: 9, page 165-184
  • ISSN: 1292-8100

Abstract

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We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a Gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical Brownian motion in some cases, by a fractional Brownian motion in other cases. The proofs of these results are based on the Lyons theory of rough paths. Finally we discuss applications in two physical situations.

How to cite

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Marty, Renaud. "Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations." ESAIM: Probability and Statistics 9 (2010): 165-184. <http://eudml.org/doc/104328>.

@article{Marty2010,
abstract = { We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a Gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical Brownian motion in some cases, by a fractional Brownian motion in other cases. The proofs of these results are based on the Lyons theory of rough paths. Finally we discuss applications in two physical situations. },
author = {Marty, Renaud},
journal = {ESAIM: Probability and Statistics},
keywords = {Limit theorems; stationary processes; rough paths.; rough paths},
language = {eng},
month = {3},
pages = {165-184},
publisher = {EDP Sciences},
title = {Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations},
url = {http://eudml.org/doc/104328},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Marty, Renaud
TI - Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 165
EP - 184
AB - We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a Gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical Brownian motion in some cases, by a fractional Brownian motion in other cases. The proofs of these results are based on the Lyons theory of rough paths. Finally we discuss applications in two physical situations.
LA - eng
KW - Limit theorems; stationary processes; rough paths.; rough paths
UR - http://eudml.org/doc/104328
ER -

References

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