On the Sn/n-Problem

The Chow-Robbins game is a classical still partly unsolved stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide if you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $$V(n,x) = \sup_{\tau }\operatorname{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$$ where $S$ is a random walk. We give a tight upper bound for $V$ when $S$ has subgassian increments. We do this by usinf the analogous time continuous problem with a standard Brownian motion as the driving process. From this we derive an easy proof for the existence of optimal stopping times in the discrete case. For the Chow-Robbins game we as well give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 10^{5}$.