We give a new relatively compact proof of the famous identity for the distribution of the first hitting time of a linear boundary by a skip-free process with stationary independent increments. The proof uses martingale identities and change of measure.
"Kendall's identity for the first crossing time revisited." Electron. Commun. Probab. 6 91 - 94, 2001. https://doi.org/10.1214/ECP.v6-1038