Multi-dimensional BSDEs whose terminal values are bounded and have bounded Malliavin derivatives

TL;DR

This paper studies multi-dimensional backward stochastic differential equations with bounded terminal values and Malliavin derivatives, providing conditions for existence and a scheme for solving quadratic BSDEs.

Contribution

It introduces an exponential integrability condition and a differential equation approach to determine solution horizons for this class of BSDEs, enabling new solution schemes.

Findings

Exponential integrability condition for BSDE solutions

Calculation of minimum solution horizon via ODE

A new scheme for solving quadratic BSDEs

Abstract

We consider a class of multi-dimensional BSDEs on a finite time horizon (containing in particular Lipschitzian-quadratic BSDEs), whose terminal values are bounded as well as their corresponding Malliavin derivatives. We prove two results. The first one is an exponential integrability condition which determines when a BSDE in this class has a solution up to a given time horizon. In the second result, via an ordinary differential equation, we compute a minimum horizon up to which any BSDE of this class has a solution. The combination of these two results leads to a new scheme to solve quadratic BSDEs.