  In this paper we establish a constructive method in order to show global
existence and regularity for a class of degenerate parabolic Cauchy problems
which satisfy a weak Hoermander condition on a subset of the domain where the
data are measurable and which have regular data on the complementary set of the
domain. This result has practical incentives related to the computation of
Greeks in reduced LIBOR market models, which are standard computable
approximations of the HJM-description of interest rate markets. The method
leads to a probabilistic scheme for the computation of the value function and
its sensitivities based on Malliavin calculus. From a practical perspective the
main contribution of the paper is an Monte-Carlo algorithm which includes
weight corrections for paths which move in time into a region where a (weak)
Hoermander condition holds.
