Second order backward stochastic differential equations under a monotonicity condition

In a recent paper, Soner, Touzi and Zhang (2012) [19] have introduced a notion of second order backward stochastic differential equations (2BSDEs), 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 yy and zz. The aim of this paper is to extend these results to the case of a generator satisfying a monotonicity condition in yy. More precisely, we prove existence and uniqueness for 2BSDEs with a generator which is Lipschitz in zz and uniformly continuous with linear growth in yy. Moreover, we emphasize throughout the paper the major difficulties and differences due to the 2BSDE framework.