A Dirichlet process characterization of a class of reflected diffusions

TL;DR

This paper characterizes a class of reflected diffusions as Dirichlet processes using a decomposition into a local martingale and a zero p-variation process, with applications to multidimensional and curved domains.

Contribution

It provides a novel Dirichlet process characterization for reflected diffusions under certain conditions, extending understanding beyond semimartingale frameworks.

Findings

Reflected diffusions can be decomposed into a local martingale and a zero p-variation process.

Reflected diffusions in polyhedral and curved domains are Dirichlet processes, not semimartingales.

The characterization applies to solutions of stochastic differential equations with reflection satisfying an ${\\mathbb{L}}^p$ continuity condition.

Abstract

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^p$ continuity condition holds with $p>1$, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum of a local martingale and a continuous, adapted process of zero $p$-variation. When $p=2$, this implies that the reflected diffusion is a Dirichlet process. Two examples are provided to motivate such a characterization. The first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks, and the second example is a family of two-dimensional reflected diffusions in curved domains. In both cases, the reflected diffusions are shown to be Dirichlet processes, but not semimartingales.