## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 4 (1972), 1293-1311.

### On Embedding Right Continuous Martingales in Brownian Motion

#### Abstract

A stopping time $T$ for the Wiener process $W(t)$ is called minimal if there is no stopping time $S \leqq T$ such that $W(S)$ and $W(T)$ have the same distribution. In the first section, it is shown that if $E\{W(T)\} = 0$, then $T$ is minimal if and only if the process $W(t \wedge T)$ is uniformly integrable. Also, if $T$ is minimal and $E\{W(T)\} = 0$ then $E\{T\} = E\{W(T)^2\}$. In the second section, these ideas are used to show that for any right continuous martingale $M(t)$, there is a right continuous family of minimal stopping times $T(t)$ such that $W(T(t))$ has the same finite joint distributions as $M(t)$. In the last section it is shown that if $T$ is defined in the manner proposed by Skorokhod (and therefore minimal) such that $W(T)$ has a stable distribution of index $\alpha > 1$ then $T$ is in the domain of attraction of a stable distribution of index $\alpha/2$.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 4 (1972), 1293-1311.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692480

**Digital Object Identifier**

doi:10.1214/aoms/1177692480

**Mathematical Reviews number (MathSciNet)**

MR343354

**Zentralblatt MATH identifier**

0267.60050

**JSTOR**

links.jstor.org

#### Citation

Monroe, Itrel. On Embedding Right Continuous Martingales in Brownian Motion. Ann. Math. Statist. 43 (1972), no. 4, 1293--1311. doi:10.1214/aoms/1177692480. https://projecteuclid.org/euclid.aoms/1177692480