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Embedding of random vectors into continuous martingales

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Dettweiler E.

Studia Mathematica

1999

tom 134

nr 3

251-268

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 ${\widetilde Ω}$ of Ω, and a filtration $({\widetilde ℱ}_t)_{t≥0}$ on ${\widetilde Ω}$, such that for every $X ∈ L^0(Ω,E)$ there is an E-valued, continuous $({\widetilde ℱ}_t)$-martingale $(M_t(X))_{t≥0}$ 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 $X ∈ L^0(Ω,ℝ)$, and for general E this leads to a representation of random vectors as stochastic integrals relative to a Brownian motion.

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1 Studia Mathematica

1 Dettweiler E.

1 1999

Wyniki wyszukiwania

Embedding of random vectors into continuous martingales