Let E be a complete, separable metric space and A be an operator on
Cbβ(E). We give an abstract definition of viscosity sub/supersolution of the
resolvent equation Ξ»uβAu=h and show that, if the comparison principle
holds, then the martingale problem for A has a unique solution. Our proofs
work also under two alternative definitions of viscosity sub/supersolution
which might be useful, in particular, in infinite dimensional spaces, for
instance to study measure-valued processes.
We prove the analogous result for stochastic processes that must satisfy
boundary conditions, modeled as solutions of constrained martingale problems.
In the case of reflecting diffusions in DβRd, our assumptions
allow D to be nonsmooth and the direction of reflection to be degenerate.
Two examples are presented: A diffusion with degenerate oblique direction of
reflection and a class of jump diffusion processes with infinite variation jump
component and possibly degenerate diffusion matrix