Embedding of random vectors into continuous martingales

Dettweiler, E.

Dettweiler, E.

Studia Mathematica, Tome 133 (1999), p. 251-268 / Harvested from The Polish Digital Mathematics Library

Résumé

Let E be a real, separable Banach space and denote by $L^{0} (Ω, E)$ the space of all E-valued random vectors defined on the probability space Ω. The following result is proved. There exists an extension $\tilde{Ω}$ of Ω, and a filtration ${({\tilde{ℱ}}_{t})}_{t \geq 0}$ on $\tilde{Ω}$ , such that for every $X \in L^{0} (Ω, E)$ there is an E-valued, continuous $({\tilde{ℱ}}_{t})$ -martingale ${(M_{t} (X))}_{t \geq 0}$ in which X is embedded in the sense that $X = M_{τ} (X)$ a.s. for an a.s. finite stopping time τ. For E = ℝ this gives a Skorokhod embedding for all $X \in L^{0} (Ω, ℝ)$ , and for general E this leads to a representation of random vectors as stochastic integrals relative to a Brownian motion.

Publié le : 1999-01-01

Zbl 0924.60017

EUDML-ID : urn:eudml:doc:216637

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     author = {E. Dettweiler},
     title = {Embedding of random vectors into continuous martingales},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {251-268},
     zbl = {0924.60017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p251bwm}
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Dettweiler, E. Embedding of random vectors into continuous martingales. Studia Mathematica, Tome 133 (1999) pp. 251-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p251bwm/

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