Quantile hedging and optimal control under stochastic target constraints

We consider the problem of finding the minimal initial data of a controlled process which 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, we show that this problem can be converted into a stochastic target problem, i.e. find the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics, and are shown to recover the explicit solution of Follmer and Leukert. We then consider the problem of miximizing a utility function under this type of quantile constraint. The previous study allows to characterize the domain in which the value fuction lies and we provide an Hamilton-Jacobi-Bellman representation of the associated value function. Contrary to standard state constraint problems, the domain is not given a-priori and we do not need to impose conditions on its boundary.