Second Order Backward Stochastic Differential Equations under Monotonicity Condition

TL;DR

This paper extends the theory of second order backward stochastic differential equations (2BSDEs) by establishing existence and uniqueness results for generators with monotonicity in y, broadening the applicability of 2BSDEs to more general conditions.

Contribution

It introduces existence and uniqueness results for 2BSDEs with generators that are Lipschitz in z and monotonic in y, under less restrictive conditions than previous works.

Findings

Proved existence and uniqueness of 2BSDEs with monotonic generators.

Extended 2BSDE theory to include generators with linear growth in y.

Highlighted key differences and challenges in the 2BSDE framework.

Abstract

In a recent paper, Soner, Touzi and Zhang [20] have introduced a notion of second order backward stochastic differential equations (2BSDEs for short), which are naturally linked to a class of fully non-linear PDEs. They proved existence and uniqueness for a generator which is uniformly Lipschitz in the variables $y$ and $z$. The aim of this paper is to extend these results to the case of a generator satisfying a monotonicity condition in $y$. More precisely, we prove existence and uniqueness for 2BSDEs with a generator which is Lipschitz in $z$ and uniformly continuous with linear growth in $y$. Moreover, we emphasize throughout the paper the major difficulties and differences due to the 2BSDE framework.