Let us assume that we observe a trajectory of a standard linear Brownian motion on the time interval [0,1].
Our aim is to stop at the moment which is in some sense the most close to the moment where the trajectory attains its ultimate maximum. This problem is very natural if we suppose that stock prices evolve as Brownian motion.
In 1999 Graversen, Peskir and Shiryaev have solved a problem of such type in there paper "Stopping Brownian Motion without Anticipation as Close as Possible to its Ultimate Maximum".
The present work deals with problems which arise in the situation considered above. In particular, the stopping time is found which is the most close to the moment where Brownian motion attains its maximum. (The measure of closeness is other than the one considered by Graversen, Peskir and Shiryaev.) The work provides explicit solutions of some similar problems for other processes.