VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (
Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in
801. For the much more difficult case of positive submartingales, see
1023 and above all Azéma,
Z. für W-theorie, 45, 1978 and its exposition
1321Keywords: Supermartingales,
Multiplicative decompositionNature: Original
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