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

TL;DR

This paper studies the regularity and boundary behavior of minimal super-solutions to constrained backward stochastic differential equations, demonstrating Lipschitz and Hölder continuity under boundedness assumptions using probabilistic methods.

Contribution

It establishes regularity properties and boundary conditions of solutions to constrained BSDEs without relying on PDE techniques, applicable to non-Markovian cases.

Findings

First component is Lipschitz in space and 1/2-Hölder in time.

Solution's path is continuous before the horizon with a face-lifted boundary condition.

The first component equals its own face-lift, confirming boundary behavior.

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 1/2-H\"older 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.