August 2021 Non-uniqueness for reflected rough differential equations
Paul Gassiat
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 57(3): 1369-1387 (August 2021). DOI: 10.1214/20-AIHP1121

Abstract

We give an example of a reflected differential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite p-variation, for some p>2). For this equation, we identify a sharp condition on the modulus of continuity of the signal under which uniqueness holds. Lévy’s modulus for Brownian motion turns out to be a boundary case. We further show that in our example, non-uniqueness holds almost surely when the driving signal is a fractional Brownian motion with Hurst index H<12. The considered equation is driven by a two-dimensional signal with one component of bounded variation, so that rough path theory is not needed to make sense of the equation.

Nous donnons un exemple d’équation différentielle avec réflexion qui peut avoir une infinité de solutions si le signal sous-jacent est suffisamment irrégulier (par exemple de p-variation infinie, pour un p>2). Pour cette équation, nous identifions une condition sur le module de continuité du signal sous laquelle il y a unicité. Le module de Lévy pour le mouvement brownien se trouve être un cas limite. Nous montrons de plus que, dans notre exemple, la non-unicité a lieu presque sûrement quand le signal est un mouvement brownien fractionnaire d’indice de Hurst H<12. L’équation que nous considérons est conduite par un signal bi-dimensionnel dont une composante est à variation bornée, on peut donc lui donner un sens san avoir besoin de la théorie des trajectoires rugueuses.

Funding Statement

This work is partially supported by the ANR via the project ANR-16-CE40-0020-01.

Acknowledgements

The author would like to thank Joseph Lehec for a helpful discussion, and Cyril Labbé for useful comments. The author is also grateful to an anonymous referee for several remarks which helped to improve the clarity of the presentation.

Citation

Download Citation

Paul Gassiat. "Non-uniqueness for reflected rough differential equations." Ann. Inst. H. Poincaré Probab. Statist. 57 (3) 1369 - 1387, August 2021. https://doi.org/10.1214/20-AIHP1121

Information

Received: 13 March 2020; Revised: 30 October 2020; Accepted: 6 November 2020; Published: August 2021
First available in Project Euclid: 22 July 2021

MathSciNet: MR4291451
zbMATH: 1480.60156
Digital Object Identifier: 10.1214/20-AIHP1121

Subjects:
Primary: 60H10 , 60L20

Keywords: fractional Brownian motion , Reflected differential equations , rough differential equations

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré

Vol.57 • No. 3 • August 2021
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