We determine $\sup E\lbrack r(S_T)\rbrack$, where $S_n$ is a sequence of partial sums of independent identically distributed random variables, for two reward functions: $r(x) = x^+$ and $r(x) = (e^x - 1)^+$. The supremum is taken over all stop rules $T$. We give conditions under which the optimal expected return is finite. Under these conditions, optimal stopping times exist, and we determine them. The problem has an interpretation in an action timing problem in finance.
D. A. Darling. T. Liggett. H. M. Taylor. "Optimal Stopping for Partial Sums." Ann. Math. Statist. 43 (4) 1363 - 1368, August, 1972. https://doi.org/10.1214/aoms/1177692491