## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 4 (1972), 1363-1368.

### Optimal Stopping for Partial Sums

D. A. Darling, T. Liggett, and H. M. Taylor

#### Abstract

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.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 4 (1972), 1363-1368.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692491

**Digital Object Identifier**

doi:10.1214/aoms/1177692491

**Mathematical Reviews number (MathSciNet)**

MR312564

**Zentralblatt MATH identifier**

0244.60037

**JSTOR**

links.jstor.org

#### Citation

Darling, D. A.; Liggett, T.; Taylor, H. M. Optimal Stopping for Partial Sums. Ann. Math. Statist. 43 (1972), no. 4, 1363--1368. doi:10.1214/aoms/1177692491. https://projecteuclid.org/euclid.aoms/1177692491