  We prove that weakly continuous solutions to martingale problems admit a
canonical regular conditional probability distribution. This allows for the
construction of time consistent convex dynamic procedures in a non dominated
setting. Making use of the martingale problem approach for continuous
diffusions and diffusions with Levy generator, we give an explicit construction
of such procedures having furthermore a Feller property. These procedures lead
to viscosity solution of fully non linear second order partial differential
equations in case of continuous diffusions. In case of diffusions with Levy
generator this provides a probabilistic approach for the resolution of non
local fully non linear second order PDE.
