  We study the optimal stopping problem of maximizing the variance of an
unkilled linear diffusion. Especially, we demonstrate how the problem can be
solved as a convex two-player zero-sum game, and reveal quite surprising
application of game theory by doing so. Our main result shows that an optimal
solution can, in general case, be found among stopping times that are mixtures
of two hitting times. This and other revealed phenomena together with suggested
solution methods could be helpful when facing more complex non-linear optimal
stopping problems. The results are illustrated by a few examples.
