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August, 1983 Valeurs Prises par les Martingales Locales Continues a un Instant Donne
M. Emery, C. Stricker, J. A. Yan
Ann. Probab. 11(3): 635-641 (August, 1983). DOI: 10.1214/aop/1176993507


By extending ideas and methods of Dudley (1977) (who was dealing with the Brownian case), we prove that a necessary and sufficient condition for all martingales of a given filtration $(\mathscr{F}_t)$ to be continuous, is that, for every stopping time $T$ and every $\mathscr{F}_T$-measurable random variable $X$, there exists a continuous local martingale $M$ with $M_T = X$ a.s. Moreover, $M$ can be chosen such that $M_0 = 0$ on a reasonably large event (equal to $\{T > 0\}$ in the Brownian case); if there exists a Brownian motion $B$ adapted to $(\mathscr{F}_t), M$ can be chosen as a stochastic integral of some $(\mathscr{F}_t)$-predictable process with respect to $B$ (even when $(\mathscr{F}_t)$ is larger than the natural filtration of $B$).


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M. Emery. C. Stricker. J. A. Yan. "Valeurs Prises par les Martingales Locales Continues a un Instant Donne." Ann. Probab. 11 (3) 635 - 641, August, 1983.


Published: August, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0517.60055
MathSciNet: MR704549
Digital Object Identifier: 10.1214/aop/1176993507

Primary: 60G44
Secondary: 60G07 , 60H05

Keywords: local martingales , predictable representation property , stochastic integrals , stopping times

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 3 • August, 1983
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