We consider a controlled process governed by $X^{x, \theta} = x +
\int \theta dS + H^{\theta}$, where $S$ is a semimartingale, $\Theta$ the set
of control processes . is a convex subset of $L(S)$ and ${H^{\theta} :\theta
\in \Theta}$ is a concave family of adapted processes with finite variation. We
study the problem of minimizing the shortfall risk defined as the expectation
of the shortfall $(B - X_T^{x, \theta})_+$ weighted by some loss function,
where $B$ is a given nonnegative measurable random variable. Such a criterion
has been introduced by Föllmer and Leukert [Finance Stoch.
4 (1999) 117–146] motivated by a hedging problem in an incomplete
financial market context:$\Theta = L(S)$ and $H^{\theta} \equiv 0$. Using
change of measures and optional decomposition under constraints, we state an
existence result to this optimization problem and show some qualitative
properties of the associated value function. A verification theorem in terms of
a dual control problem is established which is used to obtain a quantitative
description of the solution. Finally, we give some applications to hedging
problems in constrained portfolios, large investor and reinsurance models.