By Bruno Bouchard, Romuald Elie & Nizar Touzi (Cahier de la Chaire n°29)
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. nd 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 nancial mathematics, and are shown to recover the explicit solution of
Follmer and Leukert.
Keywords: Stochastic target problem, discontinuous viscosity solutions, quantile hedging