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.
Highlights
Let A be an operator determining a Markov process X with state space E as the solution of the martingale problem in which tMf (t) = f (X(t)) − f (X(0)) − Af (X(s))ds (1.1)is required to be a martingale with respect to a filtration {Ft} for all f ∈ D(A), the domain of A
An alternative approach by Stroock and Varadhan [31] introduces a submartingale problem which weakens the restriction on the domain of A to the requirement that Bf (x) ≥ 0 for x ∈ ∂E0 and requires that for all such f ∈ D(A), (1.1) is a submartingale
The primary goal of this paper is to prove a Markov selection theorem for solutions of constrained martingale problems
Summary
Is required to be a martingale with respect to a filtration {Ft} for all f ∈ D(A), the domain of A. In this case, [10] have shown that these conditions are necessary for existence of a semimartingale reflecting Brownian motion.
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