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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 theorem
Nature: Original
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