Electronic Journal of Probability

Viscosity methods giving uniqueness for martingale problems

Cristina Costantini and Thomas Kurtz

Full-text: Open access

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 $\lambda u-Au=h$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067173

Digital Object Identifier
doi:10.1214/EJP.v20-3624

Mathematical Reviews number (MathSciNet)
MR3361255

Zentralblatt MATH identifier
1341.60030

Subjects
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}

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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