Necessary and sufficient conditions for the $r$-excessive local martingales to be martingales

TL;DR

This paper establishes conditions under which certain $r$-excessive local martingales associated with Markov processes are true martingales, focusing on boundary accessibility and entrance properties.

Contribution

It provides necessary and sufficient conditions linking boundary characteristics to the martingale property of $r$-excessive local martingales.

Findings

Local martingale is strict if boundary is inaccessible and entrance.

Local martingale is a true martingale otherwise.

Conditions depend on boundary accessibility and entrance properties.

Abstract

We consider the decreasing and the increasing $r$-excessive functions $\varphi_r$ and $\psi_r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta$. We prove that the $r$-excessive local martingale $\bigl( e^{-r (t \wedge T_\alpha)} \varphi_r (X_{t \wedge T_\alpha}) \bigr)$ $\bigl($resp., $\bigl( e^{-r (t \wedge T_\beta)} \psi_r (X_{t \wedge T_\beta}) \bigr) \bigr)$ is a strict local martingale if the boundary point $\alpha$ (resp., $\beta$) is inaccessible and entrance, and a martingale otherwise.