Before revision:  We propose to study this problem in the framework of second order backward stochastic differential equations introduced in Soner, Touzi and Zhang (2010) for Lipschitz continuous generator, then generalized by Possamai and Zhou (2011) in the quadratic growth case.
After revision:  We propose to study this problem in the framework of second-order backward stochastic differential equations introduced in Soner, Touzi and Zhang (2010) for Lipschitz continuous generator, then generalized by Possamai and Zhou (2011) in the quadratic growth case.