Before revision:  Moreover, we establish the equivalence: the stochastic solution is the unique classical solution to the Cauchy problem if, and only if, a comparison theorem holds. For the case where the stochastic solution may not be smooth, we characterize it as a limit of smooth  stochastic solutions associated with some approximating Cauchy problems .
After revision:  Moreover, we establish the equivalence: the stochastic solution is not smooth stochastic solutions associated with some approximating Cauchy problems .