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2020 Solving mean field rough differential equations
Ismaël Bailleul, Rémi Catellier, François Delarue
Electron. J. Probab. 25: 1-51 (2020). DOI: 10.1214/19-EJP409

Abstract

We provide in this work a robust solution theory for random rough differential equations of mean field type \[ dX_{t} = V\big ( X_{t},{\mathcal L}(X_{t})\big )dt + \textrm{F} \bigl ( X_{t},{\mathcal L}(X_{t})\bigr ) dW_{t}, \] where $W$ is a random rough path and ${\mathcal L}(X_{t})$ stands for the law of $X_{t}$, with mean field interaction in both the drift and diffusivity. We show that, in addition to the enhanced path of $W$, the underlying rough path-like setting should also comprise an infinite dimensional component obtained by regarding the collection of realizations of $W$ as a deterministic trajectory with values in some $L^{q}$ space. This advocates for a suitable notion of controlled path à la Gubinelli inspired from Lions’ approach to differential calculus on Wasserstein space, the systematic use of the latter playing a fundamental role in our study. Whilst elucidating the rough set-up is a key step in the analysis, solving the mean field rough equation requires another effort: the equation cannot be dealt with as a mere rough differential equation driven by a possibly infinite dimensional rough path. Because of the mean field component, the proof of existence and uniqueness indeed asks for a specific and quite elaborated localization-in-time argument.

Citation

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Ismaël Bailleul. Rémi Catellier. François Delarue. "Solving mean field rough differential equations." Electron. J. Probab. 25 1 - 51, 2020. https://doi.org/10.1214/19-EJP409

Information

Received: 23 November 2019; Accepted: 23 December 2019; Published: 2020
First available in Project Euclid: 7 February 2020

zbMATH: 07206358
MathSciNet: MR4073682
Digital Object Identifier: 10.1214/19-EJP409

Subjects:
Primary: 60G99, 60H10

JOURNAL ARTICLE
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