Regularity of BSDEs with a convex constraint on the gains-process. (English) Zbl 1426.60068

Bouchard, Bruno; Elie, Romuald; Moreau, Ludovic

Bernoulli 24, No. 3, 1613-1635 (2018).

Summary: 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 \(\frac{1}{2}\)-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.

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

backward stochastic differential equation with a constraint; regularity; stability