| Zusammenfassung: | We consider a controlled process governed by Xx,θ= x + ∫θ dS + Hθ, where S is a semimartingale, Θ the set of control processes θ is a convex subset of L(S) and Hθ: θ∈Θ 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 - Xx, θ T)+ weighted by some loss function, where B is a given nonnegative measurable random variable. Such a criterion has been introduced by Follmer and Leukert [Finance Stoch. 4 (1999) 117-146] motivated by a hedging problem in an incomplete financial market context: Θ = 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.
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