  We extend the study of [7, 18] to stochastic target problems with general
market impacts. Namely, we consider a general abstract model which can be
associated to a fully nonlinear parabolic equation. Unlike [7, 18], the
equation is not concave and the regularization/verification approach of [7] can
not be applied. We also relax the gamma constraint of [7]. In place, we need to
generalize the a priori estimates of [18] and exhibit smooth solutions from the
classical parabolic equations theory. Up to an additional approximating
argument, this allows us to show that the super-hedging price solves the
parabolic equation and that a perfect hedging strategy can be constructed when
the coefficients are smooth enough. This representation leads to a general dual
formulation. We finally provide an asymptotic expansion around a model without
impact.
