Electronic Journal of Probability

Viscosity methods giving uniqueness for martingale problems

Cristina Costantini and Thomas Kurtz

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 67, 27 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60G46: Martingales and classical analysis 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

martingale problems uniqueness metric space viscosity solution boundary conditions constrained martingale problem

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Costantini, Cristina; Kurtz, Thomas. Viscosity methods giving uniqueness for martingale problems. Electron. J. Probab. 20 (2015), paper no. 67, 27 pp. doi:10.1214/EJP.v20-3624. https://projecteuclid.org/euclid.ejp/1465067173

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