On predicting the ultimate maximum for exponential Lévy processes (Q456278)

The authors take exponential Lévy process and consider the probelm of optimal stopping in the interval \([0,T]\) of the ratio of the process itself and its maximal value, and vice versa. More precisely, \(\alpha\)-stable and generalized hyperbolic Lévy processes are studied. This problem corresponds to the linear utility, however, logarithmic utility is considered as well. The optimal solution is trivial, in some sense: it equals the maturity date otherwise it equals the initial date. The advantage of the proofs is that they do not depend on the particular change of time for the underlying process. The application to the selling of assets is discussed.