## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 3 (1984), 876-881.

### Approximations to Optimal Stopping Rules for Exponential Random Variables

#### Abstract

For $X_1, X_2, \cdots$ i.i.d. with finite mean and $Y_n = \max(X_1, \cdots, X_n) - cn, c$ positive, a number of authors have considered the problem of determining an optimal stopping rule for the reward sequence $Y_n$. The optimal stopping rule can be given explicitly in this case; however, in general its use requires complete knowledge of the distribution of the $X_i$. This paper examines the problem of approximating the optimal expected reward when only partial information about the distribution is available. Specifically, if the $X_i$ are known to be exponentially distributed with unknown mean, stopping rules designed to approximate the optimal rule (which can be used only when the mean is known) are proposed. Under certain conditions the difference between the expected reward using the proposed stopping rules and the optimal expected reward vanishes as $c$ approaches zero.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 3 (1984), 876-881.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993236

**Digital Object Identifier**

doi:10.1214/aop/1176993236

**Mathematical Reviews number (MathSciNet)**

MR744242

**Zentralblatt MATH identifier**

0544.62076

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L15: Optimal stopping [See also 60G40, 91A60]

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Optimal stopping uniform integrability last times

#### Citation

Martinsek, Adam T. Approximations to Optimal Stopping Rules for Exponential Random Variables. Ann. Probab. 12 (1984), no. 3, 876--881. doi:10.1214/aop/1176993236. https://projecteuclid.org/euclid.aop/1176993236