# Markov selection for constrained martingale problems

## TL;DR

This paper develops a method to select strong Markov solutions for constrained martingale problems, including reflecting diffusions, ensuring uniqueness and providing tools for analyzing boundary behaviors in stochastic processes.

## Contribution

It introduces a Markov selection approach for constrained martingale problems, extending existence and uniqueness results to more general reflecting diffusions and boundary conditions.

## Key findings

- Constructed strong Markov solutions for constrained processes.
- Proved uniqueness among strong Markov solutions implies uniqueness overall.
- Applied results to reflecting diffusions in complex domains.

## Abstract

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 Dupuis and Ishii (1993): In fact our conditions are known to be optimal in the case of simple, convex polyhedrons with constant direction of reflection on each face (Dai and Williams (1995)). We also indicate how the results can be applied to processes with Wentzell boundary conditions and nonlocal boundary conditions.