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2015 Viscosity methods giving uniqueness for martingale problems
Cristina Costantini, Thomas Kurtz
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Electron. J. Probab. 20: 1-27 (2015). DOI: 10.1214/EJP.v20-3624


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 $D\subset {\bf R}^d$, 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.


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Cristina Costantini. Thomas Kurtz. "Viscosity methods giving uniqueness for martingale problems." Electron. J. Probab. 20 1 - 27, 2015.


Accepted: 17 June 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1341.60030
MathSciNet: MR3361255
Digital Object Identifier: 10.1214/EJP.v20-3624

Primary: 60J25
Secondary: 47D07, 60G46, 60J35


Vol.20 • 2015
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