A class of optimal stopping problems for the Wiener process is studied herein, and asymptotic expansions for the optimal stopping boundaries are derived. These results lead to a simple index-type class of asymptotically optimal solutions to the classical discounted multi-armed bandit problem: given a discount factor 0<β <1 and k populations with densities from an exponential family, how should x1, x2,… be sampled sequentially from these populations to maximize the expected value of Ʃ∞1 βi−1xi, in ignorance of the parameters of the densities?
