Stochastic Target Problems with Controlled Loss
This is a joint work with Romuald Elie and Bruno Bouchard. 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.
Nizar Touzi, Imperial College London, United Kingdom and Ecole Polytechnique,
France
