Sequential testing problems for Lévy processes

We present the sequential testing of two simple hypotheses for a large class of Lévy processes. As usual in this framework, the initial optimal stopping problem is reduced to a free-boundary problem, solved through the principles of the smooth and/or continuous fit. The well-known solutions of the Wiener and the Poisson sequential testing can be derived from our procedure. The exact solution for sequentially testing two simple hypotheses concerning the parameter p, 0 < p < 1, of a negative binomial process is explicitly given.