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Solving mean field rough differential equations

Abstract : We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ 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. The analysis requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way.
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Submitted on : Wednesday, February 26, 2020 - 2:59:26 PM
Last modification on : Thursday, January 20, 2022 - 9:02:01 AM

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Ismaël Bailleul, Rémi Catellier, F. Delarue. Solving mean field rough differential equations. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2020, 25, paper no. 21, 51 pp. ⟨10.1214/19-EJP409⟩. ⟨hal-02491950⟩



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