On the Feller-Dynkin and the Martingale Property of One-Dimensional Diffusions

TL;DR

This paper establishes a precise condition under which one-dimensional regular continuous Markov processes are Feller--Dynkin processes, linking the Feller--Dynkin property to the martingale property of the scale-transformed process, and discusses implications for diffusions.

Contribution

It proves that for one-dimensional regular diffusions, the Feller--Dynkin property is equivalent to the martingale property of the scale function process, and highlights the failure of this equivalence in higher dimensions.

Findings

Feller--Dynkin and martingale properties are equivalent for 1D regular diffusions on natural scale.

Counterexample shows the equivalence does not extend to multi-dimensional diffusions.

Discussion of relations to Cauchy problems for Itô diffusions.

Abstract

We show that a one-dimensional regular continuous Markov process \(\X\) with scale function \(s\) is a Feller--Dynkin process precisely if the space transformed process \(s (X)\) is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller--Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. By means of a counterexample, we also show that this equivalence fails for multi-dimensional diffusions. Moreover, for It\^o diffusions we discuss relations to Cauchy problems.