Geodesic
On a sufficient condition for a diffusion process will nether reach boundaries of some interval
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 291-304

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A one-dimensional diffusion semi-Markov process on some interval of its values is considered. Semi-Markov transition functions of the process dstisfy a second order differential equation with coefficients admitting possibility what the process stops inside this interval. In terms of coefficients of this equation some sufficient conditions are proved for the process will nether reach the left or right boundaries of this interval. 
@article{ZNSL_2020_495_a17,
     author = {B. P. Harlamov},
     title = {On a sufficient condition for a diffusion process will nether reach boundaries of some interval},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {291--304},
     publisher = {mathdoc},
     volume = {495},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a17/}
}
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B. P. Harlamov. On a sufficient condition for a diffusion process will nether reach boundaries of some interval. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 291-304. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a17/