## The Annals of Mathematical Statistics

### On Embedding Right Continuous Martingales in Brownian Motion

Itrel Monroe

#### 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

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