Bounds for Stopped Partial Sums

Abstract

Upper bounds are derived for the expected value of a stopped random sum under each of four sets of assumptions concerning the summands, plus under an additional set describing a related and similar problem. Too complex to abstract, these assumptions in part, typically limit the first two moments of the summands. The bounds have an interpretation in a stock market timing problem in which the random sum represents the sequence of daily prices of a stock and the positive part of the sum reflects a potential profit a holder of an option in the stock could realize were he to exercise. In only one case are the summands required to be independent and identically distributed, and thus we obtain bounds on the expected profit that do not require the controversial random walk model for stock prices. Of course, the bounds are of interest for other reasons as well. For example, as a related result we show that if $(S_n)$ is a random walk for which the summands $(X_n)$ have a negative mean, then $E\lbrack S_T^+\rbrack < \infty$ for all stopping times $T$ if and only if $E\lbrack (X_1^+)^2\rbrack < \infty$. For the most part, techniques familiar to readers of Dubins and Savage (How to Gamble if You Must, McGraw-Hill 1965) are used.