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. We solve the problem for exponential, power and logarithmic utility functions and prove existence of an optimal strategy.
After revision:  We propose to study this problem in the framework of second order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utility functions and prove existence of an optimal strategy.