A rather complicated condition is shown to be necessary and sufficient for a Lévy process to have points of increase. A much simpler condition is then shown to be sufficient in the general case, and necessary under certain regularity conditions. The approach used here also gives a unified proof of results for certain special classes of Lévy processes, which have previously been obtained by Bertoin.
"Increase of Lévy processes." Ann. Probab. 24 (2) 961 - 970, April 1996. https://doi.org/10.1214/aop/1039639372