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Viscosity methods giving uniqueness for martingale problems. (English) Zbl 1341.60030
Summary: 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 $$\lambda u-Au=h$$ 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 $$D\subset \mathbb{R}^d$$, 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.

##### MSC:
 60G44 Martingales with continuous parameter 60J60 Diffusion processes 60J75 Jump processes (MSC2010) 60J25 Continuous-time Markov processes on general state spaces 60J35 Transition functions, generators and resolvents 60G46 Martingales and classical analysis 47D07 Markov semigroups and applications to diffusion processes
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