## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 1613-1635.

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

Bruno Bouchard, Romuald Elie, and Ludovic Moreau

#### Abstract

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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 1613-1635.

**Dates**

Received: September 2014

Revised: June 2015

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540455

**Digital Object Identifier**

doi:10.3150/16-BEJ806

**Mathematical Reviews number (MathSciNet)**

MR3757510

**Zentralblatt MATH identifier**

06839247

**Keywords**

backward stochastic differential equation with a constraint regularity stability

#### Citation

Bouchard, Bruno; Elie, Romuald; Moreau, Ludovic. Regularity of BSDEs with a convex constraint on the gains-process. Bernoulli 24 (2018), no. 3, 1613--1635. doi:10.3150/16-BEJ806. https://projecteuclid.org/euclid.bj/1517540455