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arXiv · 2101.02002

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

David Criens
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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.

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David Criens. 2021-01-06. On the Feller-Dynkin and the Martingale Property of One-Dimensional Diffusions. https://arxiv.org/abs/2101.02002

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