On optimal stopping of a sequence of independent random variables -- probability maximizing approach

Let [xi]1,[xi]2,... be a sequence of independent, identically distributed r.v. with a continuous distribution function. Optimal stopping problems are considered for: (1) a finite sequence [xi]1,...,[xi]N, (2) sequences ([xi]n-cn)n[epsilon]N and (max([xi]1,...,[xi]n) - cn)n[epsilon]N, where c is a fixed positive number, (3) the sequence ([xi]n)n[epsilon]N, where it is additionally assumed that [xi]1,[xi]2,... appear according to a Poisson process which is independent of {[xi]n}n[epsilon]N, and the decision about stopping must be made before some fixed moment T. The object of optimization is not (as it is in the classical formulation of optimal stopping problems) the expected value of the reward, but the probability that at the moment of stopping the reward attains its maximal value. It is proved that optimal stopping rules (in the above sense) for all problems exist, and their forms are found.