  Since the work of Page in the 1950s, the problem of detecting an abrupt
change in the distribution of stochastic processes has received a great deal of
attention. In particular, a deep connection has been established between
Lorden's minimax approach to change-point detection and the widely used CUSUM
procedure, first for discrete-time processes, and subsequently for some of
their continuous-time counterparts. However, results for processes with jumps
are still scarce, while the practical importance of such processes has
escalated since the turn of the century. In this work we consider the problem
of detecting a change in the distribution of continuous-time processes with
independent and stationary increments, i.e. L\'evy processes, and our main
result shows that CUSUM is indeed optimal in Lorden's sense. This is the most
natural continuous-time analogue of the seminal work of Moustakides for
sequentially observed random variables that are assumed to be i.i.d. before and
after the change-point. From a practical perspective, the approach we adopt is
appealing as it consists in approximating the continuous-time problem by a
suitable sequence of change-point problems with equispaced sampling points, and
for which a CUSUM procedure is shown to be optimal.
