The non-linear sewing lemma I : weak formulation

Antoine Brault (1, 2) , Antoine Lejay (3, 4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 TOSCA - TO Simulate and CAlibrate stochastic models
4 IECL - Institut Élie Cartan de Lorraine

We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.

Mots clés: measurable flows; non uniqueness of solutions; rough paths; rough differential equations; Lipschitz flows; sewing lemma; flow approximations

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Antoine Brault, Antoine Lejay. The non-linear sewing lemma I : weak formulation. Electronic Journal of Probability, In press, 24 (59), pp.1-24. ⟨10.1214/19-EJP313⟩. ⟨hal-01716945v5⟩