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2019 Markov selection for constrained martingale problems
Cristina Costantini, Thomas G. Kurtz
Electron. J. Probab. 24: 1-31 (2019). DOI: 10.1214/19-EJP393


Constrained Markov processes, such as reflecting diffusions, behave as an unconstrained process in the interior of a domain but upon reaching the boundary are controlled in some way so that they do not leave the closure of the domain. In this paper, the behavior in the interior is specified by a generator of a Markov process, and the constraints are specified by a controlled generator. Together, the generators define a constrained martingale problem. The desired constrained processes are constructed by first solving a simpler controlled martingale problem and then obtaining the desired process as a time-change of the controlled process. As for ordinary martingale problems, it is rarely obvious that the process constructed in this manner is unique. The primary goal of the paper is to show that from among the processes constructed in this way one can “select”, in the sense of Krylov, a strong Markov process. Corollaries to these constructions include the observation that uniqueness among strong Markov solutions implies uniqueness among all solutions. These results provide useful tools for proving uniqueness for constrained processes including reflecting diffusions. The constructions also yield viscosity semisolutions of the resolvent equation and, if uniqueness holds, a viscosity solution, without proving a comparison principle. We illustrate our results by applying them to reflecting diffusions in piecewise smooth domains. We prove existence of a strong Markov solution to the SDE with reflection, under conditions more general than in [13]: In fact our conditions are known to be optimal in the case of simple, convex polyhedrons with constant direction of reflection on each face ([10]). We also indicate how the results can be applied to processes with Wentzell boundary conditions and nonlocal boundary conditions.


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Cristina Costantini. Thomas G. Kurtz. "Markov selection for constrained martingale problems." Electron. J. Probab. 24 1 - 31, 2019.


Received: 30 January 2019; Accepted: 7 November 2019; Published: 2019
First available in Project Euclid: 14 November 2019

zbMATH: 07142929
MathSciNet: MR4040995
Digital Object Identifier: 10.1214/19-EJP393

Primary: 60J25 , 60J50
Secondary: 60H30 , 60J60

Keywords: boundary control , constrained martingale problems , Markov selection , nonlocal boundary conditions , reflecting diffusion , viscosity solution , Wentzell boundary conditions


Vol.24 • 2019
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