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We extend the study of [7, 18] to stochastic target problems with generalmarket impacts. Namely, we consider a general abstract model which can beassociated to a fully nonlinear parabolic equation. Unlike [7, 18], theequation is not concave and the regularization/verification approach of [7] cannot be applied. We also relax the gamma constraint of [7]. In place, we need togeneralize the a priori estimates of [18] and exhibit smooth solutions from theclassical parabolic equations theory. Up to an additional approximatingargument, this allows us to show that the super-hedging price solves theparabolic equation and that a perfect hedging strategy can be constructed whenthe coefficients are smooth enough. This representation leads to a general dualformulation. We finally provide an asymptotic expansion around a model withoutimpact.

Second order stochastic target problems with generalized market impact