An optimal stopping problem with finite horizon for sums of i.i.d. random variables (Q1098165)

For a sequence \(U_ 1,U_ 2,..\). of i.i.d. random variables with zero expectation and unit variance, the optimal stopping problem for \((n- j)^{\ell}(U_ 1+...+U_ j)\), \(1\leq j\leq n\), \(\ell\) some fixed positive integer, is treated. Using backward induction, the general structure of the optimal stopping time is obtained. Furthermore, by considering the corresponding stopping problem for Brownian motion which may be explicitly solved using a time-change argument, the author derives the asymptotic behaviour of the optimal stopping time as n tends to infinity.