The Annals of Mathematical Statistics

On Embedding Right Continuous Martingales in Brownian Motion

Itrel Monroe

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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


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