Viscosity methods giving uniqueness for martingale problems

Abstract

Let E be a complete, separable metric space and A be an operator on C b (E). We give an abstract definition of viscosity sub/supersolution of the resolvent equation λu − Au = 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 D ⊂ R d, 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.