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XIII: 58, 642-645, LNM 721 (1979)
MAISONNEUVE, Bernard
Martingales de valeur absolue donnée, d'après Protter-Sharpe (Martingale theory)
The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (Ann. Prob., 7, 1979). This proof is further simplified and slightly generalized
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