Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Differential equations driven by Gaussian signals

Peter Friz and Nicolas Victoir

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Abstract

We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful – yet conceptually simple – framework in which to analyze differential equations driven by Gaussian signals in the rough paths sense.

Résumé

Nous donnons une condition simple et optimale sur la covariance d’un processus gaussien pour que celui-ci puisse être associé naturellement à un rough path. Une fois ce processus construit, nous démontrons un principe de grandes déviations, un théorème du support, et plusieurs résultats d’approximations. Avec la théorie des rough paths de T. Lyons, nous obtenons ainsi un cadre puissant, bien que conceptuellement simple, dans lequel nous pouvons analyser les équations différentielles conduites par des signaux gaussiens dans le sens des rough paths.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2 (2010), 369-413.

Dates
First available in Project Euclid: 11 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1273584128

Digital Object Identifier
doi:10.1214/09-AIHP202

Mathematical Reviews number (MathSciNet)
MR2667703

Zentralblatt MATH identifier
1202.60058

Subjects
Primary: 60G15: Gaussian processes 60H99: None of the above, but in this section

Keywords
Rough paths Gaussian processes

Citation

Friz, Peter; Victoir, Nicolas. Differential equations driven by Gaussian signals. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 369--413. doi:10.1214/09-AIHP202. https://projecteuclid.org/euclid.aihp/1273584128


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