Discrete Time Optimal Stopping Problems with Fractional Rewards

In this paper, to the first, we give the general formulation of a discrete time optimal stopping problem under the criterion of a fractional reward. Using the standard mathematical argument in the theory of mathematical programming, we prove the existence of optimal stopping times for this problem. Furthermore we discuss f-optimal stopping times, finite stopping times, and Markov case, respectively. To the next, we give the formulation of new stopping problems which combined the discrete time multiple stopping problem and the criterion of a fractional reward, and then prove the existence of optimal stopping times for this problem.