XIV: 07, 62-75, LNM 784 (1980)
BARLOW, Martin T.;
YOR, Marc
Sur la construction d'une martingale continue de valeur absolue donnée (
Martingale theory)
This paper consists of two notes on Gilat's theorem (
Ann. Prob. 5, 1977, See also
1358). The problem consists in constructing, given a continuous positive submartingale $Y$, a
continuous martingale $X$ (possibly on a different space) such that $|X|$ has the same law as $Y$. Let $A$ be the increasing process associated with $Y$; it is necessary for the existence of $X$ that $dA$ should be carried by $\{Y=0\}$. This is shown by the first note (Yor's) to be also sufficient---more precisely, in this case the solutions of Gilat's problem are all continuous. The second note (Barlow's) shows how to construct a Gilat martingale by ``putting a random $\pm$ sign in front of each excursion of $Y$'', a simple intuitive idea and a delicate proof
Keywords: Gilat's theoremNature: Original
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