  In this paper we give solution to the quickest drift change detection problem
for a L\'evy process consisting of both a continuous Gaussian part and a jump
component. We consider here Bayesian framework with an exponential a priori
distribution of the change point using an optimality criterion based on a
probability of false alarm and an expected delay of the detection. Our approach
is based on the optimal stopping theory and solving some boundary value
problem. Paper is supplemented by an extensive numerical analysis related with
the construction of the Generalized Shiryaev-Roberts statistics. In particular,
we apply this method (after appropriate calibration) to analyse Polish life
tables and to model the force of mortality in this population with a drift
changing in time.
