Before revision:  An advantage with our approach is that it also gives\emph{ [  an explicit relation between the optimal portfolio \varphi^* and the optimal measure Q^* , in terms of backward stochastic differential equations .
After revision:  An advantage with our approach is that it also gives\emph{ [ . We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, 0 \leq t \leq T. In the terminal time case t=T we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \varphi^* and the optimal measure Q^* , in terms of backward stochastic differential equations .