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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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