Minimizing Shortfall Risk and Applications to Finance and Insurance Problems

We consider a controlled process governed by Xx,.=x+..dS+H., where S is a semimartingale, . the set of control processes . is a convex subset of L(S) and H.:... 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,.T)+ 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:.=L(S) and H..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.