We consider spectrally negative Lévy process and determine the joint Laplace transform of the exit time and exit position from an interval containing the origin of the process reflected in its supremum. In the literature of fluid models, this stopping time can be identified as the time to buffer-overflow. The Laplace transform is determined in terms of the scale functions that appear in the two-sided exit problem of the given Lévy process. The obtained results together with existing results on two sided exit problems are applied to solving optimal stopping problems associated with the pricing of Russian options and their Canadized versions.
"Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options." Ann. Appl. Probab. 14 (1) 215 - 238, February 2004. https://doi.org/10.1214/aoap/1075828052