Abstract (EN)

We consider the problem of finding the minimal initial data of a controlled processwhich guarantees to reach a controlled target with a given probability of success or, more generally,with a given level of expected loss. By suitably increasing the state space and the controls, weshow that this problem can be converted into a stochastic target problem, i.e., finding the minimalinitial data of a controlled process which guarantees to reach a controlled target with probability one.Unlike in the existing literature on stochastic target problems, our increased controls are valued in anunbounded set. In this paper, we provide a new derivation of the dynamic programming equation forgeneral stochastic target problems with unbounded controls, together with the appropriate boundaryconditions. These results are applied to the problem of quantile hedging in financial mathematicsand are shown to recover the explicit solution of F¨ ollmer and Leukert [Finance Stoch., 3 (1999), pp.251–273].