# Embedding of random vectors into continuous martingales

Studia Mathematica (1999)

- Volume: 134, Issue: 3, page 251-268
- ISSN: 0039-3223

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topDettweiler, E.. "Embedding of random vectors into continuous martingales." Studia Mathematica 134.3 (1999): 251-268. <http://eudml.org/doc/216637>.

@article{Dettweiler1999,

abstract = {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.},

author = {Dettweiler, E.},

journal = {Studia Mathematica},

keywords = {Skorokhod embedding; martingale; stochastic integral; Brownian motion; stopping time},

language = {eng},

number = {3},

pages = {251-268},

title = {Embedding of random vectors into continuous martingales},

url = {http://eudml.org/doc/216637},

volume = {134},

year = {1999},

}

TY - JOUR

AU - Dettweiler, E.

TI - Embedding of random vectors into continuous martingales

JO - Studia Mathematica

PY - 1999

VL - 134

IS - 3

SP - 251

EP - 268

AB - 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.

LA - eng

KW - Skorokhod embedding; martingale; stochastic integral; Brownian motion; stopping time

UR - http://eudml.org/doc/216637

ER -

## References

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