Abstract
We show that a one-dimensional regular continuous Markov process with scale function is a Feller–Dynkin process precisely if the space transformed process 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 multidimensional diffusions. Moreover, for Itô diffusions we discuss relations to Cauchy problems.
Funding Statement
Financial support from the DFG project No. SCHM 2160/15-1 is gratefully acknowledged.
Acknowledgments
The author is very grateful to two anonymous referees for many helpful comments.
Citation
David Criens. "On the Feller–Dynkin and the martingale property of one-dimensional diffusions." Electron. Commun. Probab. 28 1 - 15, 2023. https://doi.org/10.1214/23-ECP524
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