Regularity of BSDEs with a convex constraint on the gains-process

Bruno Bouchard (1, 2) , Romuald Elie (3) , Ludovic Moreau (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CREST - Centre de Recherche en Économie et Statistique
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
4 D-MATH - Department of Mathematics [ETH Zurich]

We consider the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process. The terminal condition is given by a function of the terminal value of a forward stochastic differential equation. Under boundedness assumptions on the coefficients, we show that the first component of the solution is Lipschitz in space and 1/2-Hölder in time with respect to the initial data of the forward process. Its path is continuous before the time horizon at which its left-limit is given by a face-lifted version of its natural boundary condition. This first component is actually equal to its own face-lift. We only use probabilistic arguments. In particular, our results can be extended to certain non-Markovian settings.

Mots clés: regularity; Backward stochastic differential equation with a constraint; stability

Domaines: Probabilités [math.PR]

Bruno Bouchard, Romuald Elie, Ludovic Moreau. Regularity of BSDEs with a convex constraint on the gains-process. Bernoulli, 2018, 24 (3), pp.1613-1635. ⟨hal-01065794⟩