Geodesic
On the submartingale/supermartingale property of diffusions in natural scale
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 129-139.

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S. Kotani (2006) has characterised the martingale property of a one-dimensional diffusion in natural scale in terms of the classification of its boundaries. We complement this result by establishing a necessary and sufficient condition for a one-dimensional diffusion in natural scale to be a submartingale or a supermartingale. Furthermore, we study the asymptotic behaviour of the diffusion's expected state at time $t$ as $t\to\infty$. We illustrate our results by means of several examples. 
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Alexander Gushchin; Mikhail Urusov; Mihail Zervos. On the submartingale/supermartingale property of diffusions in natural scale. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 129-139. http://geodesic.mathdoc.fr/item/TM_2014_287_a7/

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