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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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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

We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i..e. in a neighborhood of the origin has the formGiven a vector field γ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin γ i (0) := lim x 1 →0 + γ(x 1 , ψ i (x 1 )), i = 1, 2, and assuming there exists a vector e * such that e * , γ i (0) > 0, i = 1, 2, and e * 1 > 0, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin. Our proof uses a new scaling result and a coupling argument.