  In a Markovian framework, we consider the problem of finding the minimal
initial value of a controlled process allowing to reach a stochastic target
with a given level of expected loss. This question arises typically in
approximate hedging problems. The solution to this problem has been
characterised by Bouchard, Elie and Touzi in [1] and is known to solve an
Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we
prove a comparison theorem for the corresponding PDE by showing first that it
can be rewritten using a continuous operator, in some cases. As an application,
we then study the quantile hedging price of Bermudan options in the non-linear
case, pursuing the study initiated in [2].
  [1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems
with controlled loss. SIAM Journal on Control and Optimization,
48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et
al. Bsdes with weak terminal condition. The Annals of Probability,
43(2):572-604,2015.
