Viscosity methods giving uniqueness for martingale problems

TL;DR

This paper introduces viscosity solution methods for martingale problems, establishing uniqueness under comparison principles, and extends the framework to infinite-dimensional and boundary-constrained processes with practical examples.

Contribution

It provides an abstract viscosity solution framework for martingale problems, including in infinite-dimensional spaces and with boundary conditions, ensuring uniqueness and broad applicability.

Findings

Viscosity solutions ensure uniqueness of martingale problem solutions.

Framework applies to infinite-dimensional and boundary-constrained processes.

Includes examples with degenerate and jump diffusions.

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.