The Asymptotics of Multiple Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density and are sequentially observed, with no recall of previous observations. The objective of such problems is to find a procedure which maximizes the expected reward. We first present a methodology for obtaining asymptotic expressions for the expectation and variance of the single optimal stopping time as the number of drawn variables becomes large. We then extend these results to the multiple stopping problem where the objective is now to maximize the expected reward, which is the sum of all variables stopped on. For a family of distributions with exponential tails and for the uniform distribution, we provide the complete generalisation of the multiple stopping problem by computing the inductive behaviour of the stopping time. Explicit calculations are provided for several common probability density functions as well as numerical simulations to support our asymptotic predictions.