Differential equations driven by rough signals.

Terry J. Lyons

Revista Matemática Iberoamericana (1998)

  • Volume: 14, Issue: 2, page 215-310
  • ISSN: 0213-2230

Abstract

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This paper aims to provide a systematic approach to the treatment of differential equations of the typedyt = Σi fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process.However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]).We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.

How to cite

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Lyons, Terry J.. "Differential equations driven by rough signals.." Revista Matemática Iberoamericana 14.2 (1998): 215-310. <http://eudml.org/doc/39555>.

@article{Lyons1998,
abstract = {This paper aims to provide a systematic approach to the treatment of differential equations of the typedyt = Σi fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process.However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]).We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.},
author = {Lyons, Terry J.},
journal = {Revista Matemática Iberoamericana},
keywords = {Ecuaciones diferenciales estocásticas; Proceso de difusión; Movimiento browniano; stochastic differential equations; rough paths; Brown motion; Gauss and Markov processes; Lie algebras},
language = {eng},
number = {2},
pages = {215-310},
title = {Differential equations driven by rough signals.},
url = {http://eudml.org/doc/39555},
volume = {14},
year = {1998},
}

TY - JOUR
AU - Lyons, Terry J.
TI - Differential equations driven by rough signals.
JO - Revista Matemática Iberoamericana
PY - 1998
VL - 14
IS - 2
SP - 215
EP - 310
AB - This paper aims to provide a systematic approach to the treatment of differential equations of the typedyt = Σi fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process.However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]).We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.
LA - eng
KW - Ecuaciones diferenciales estocásticas; Proceso de difusión; Movimiento browniano; stochastic differential equations; rough paths; Brown motion; Gauss and Markov processes; Lie algebras
UR - http://eudml.org/doc/39555
ER -

Citations in EuDML Documents

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  1. Xiang-Dong Li, Terry J. Lyons, Smoothness of Itô maps and diffusion processes on path spaces (I)
  2. Roger Züst, Some Results on Maps That Factor through a Tree
  3. Raluca M. Balan, Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise
  4. Raluca M. Balan, -theory for the stochastic heat equation with infinite-dimensional fractional noise
  5. A. Neuenkirch, I. Nourdin, A. Rößler, S. Tindel, Trees and asymptotic expansions for fractional stochastic differential equations
  6. Renaud Marty, Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
  7. Renaud Marty, Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
  8. Terry Lyons, Nicolas Victoir, An extension theorem to rough paths
  9. Annie Millet, Marta Sanz-Solé, Large deviations for rough paths of the fractional brownian motion
  10. Peter Friz, Sebastian Riedel, Convergence rates for the full gaussian rough paths

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