BSDEs with mean reflection

Philippe Briand (1) , Romuald Elie (2) , Ying Hu (3)

1 LAMA - Laboratoire de Mathématiques
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we study a new type of BSDE, where the distribution of the Y-component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K) with deterministic K, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on Y. In particular, we provide an application to the super hedging of claims under running risk management constraint.

Mots clés: Skorokhod type minimal condition; Super-hedging; risk management constraint; Backward stochastic differential equation; mean reflection

Domaines: Probabilités [math.PR]; Gestion des risques [q-fin.RM]

Philippe Briand, Romuald Elie, Ying Hu. BSDEs with mean reflection. The Annals of Applied Probability, 2018, 28 (1), pp.482-510. ⟨10.1214/17-AAP1310⟩. ⟨hal-01318649⟩