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August, 1972 On Embedding Right Continuous Martingales in Brownian Motion
Itrel Monroe
Ann. Math. Statist. 43(4): 1293-1311 (August, 1972). DOI: 10.1214/aoms/1177692480


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 $\alpha/2$.


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Itrel Monroe. "On Embedding Right Continuous Martingales in Brownian Motion." Ann. Math. Statist. 43 (4) 1293 - 1311, August, 1972.


Published: August, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0267.60050
MathSciNet: MR343354
Digital Object Identifier: 10.1214/aoms/1177692480

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 4 • August, 1972
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