## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 4 (1981), 642-647.

### Transitivity in Problems of Optimal Stopping

#### Abstract

In a sequential decision problem it is usually assumed that the available information is represented by an increasing family $\mathscr{F}$ of $\sigma$-algebras. Often a reduction, e.g., according to principles of sufficiency or invariance, is performed which yields a smaller family $\mathscr{G}$. The consequences of such a reduction for problems of optimal stopping are treated in this paper. It is shown that $\mathscr{G}$ is transitive for $\mathscr{F}$ (in the Bahadur sense) if and only if for any stochastic process adapted to $\mathscr{G}$ the value (i.e., maximal reward by optimal stopping) under $\mathscr{G}$ and the value under $\mathscr{F}$ are equal.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 4 (1981), 642-647.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994369

**Digital Object Identifier**

doi:10.1214/aop/1176994369

**Mathematical Reviews number (MathSciNet)**

MR624690

**Zentralblatt MATH identifier**

0469.60039

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Stopping times optimal stopping transitivity

#### Citation

Irle, Albrecht. Transitivity in Problems of Optimal Stopping. Ann. Probab. 9 (1981), no. 4, 642--647. doi:10.1214/aop/1176994369. https://projecteuclid.org/euclid.aop/1176994369