Embedding of random vectors into continuous martingales

E. Dettweiler

Studia Mathematica (1999)

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

Abstract

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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 Ω ˜ of Ω, and a filtration ( ℱ ˜ t ) t ≥ 0 on Ω ˜ , such that for every X ∈ L 0 ( Ω , E ) there is an E-valued, continuous ( 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.

How to cite

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Dettweiler, 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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  1. [1] J. Azéma et M. Yor, Une solution simple au problème de Skorohod, in: Séminaire de Probabilités XIII, Lecture Notes in Math. 721, Springer, 1979, 90-115 and 625-633. Zbl0414.60055
  2. [2] R. Chacon and J. B. Walsh, One-dimensional potential embedding, in: Séminaire de Probabilités X, Lecture Notes in Math. 511, Springer, 1976, 19-23. Zbl0329.60041
  3. [3] E. Dettweiler, Banach space valued processes with independent increments and stochastic integration, in: Probability in Banach Spaces IV, Lecture Notes in Math. 990, Springer, 1983, 54-83. Zbl0514.60010
  4. [4] E. Dettweiler, Stochastic integration of Banach space valued functions, in: L. Arnold and P. Kotelenez (eds.), Stochastic Space-Time Models and Limit Theorems, Reidel, 1985, 53-79. 
  5. [5] L. Dubins, On a theorem of Skorohod, Ann. Math. Statist. 39 (1968), 2094-2097. Zbl0185.45103
  6. [6] L. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916. Zbl0203.17504
  7. [7] R. M. Dudley, Wiener functionals as Itô integrals, Ann. Probab. 5 (1977), 140-141. Zbl0359.60071
  8. [8] R. M. Dudley, Real Analysis and Probability, Wadsworth & Brooks-Cole, 1989. Zbl0686.60001
  9. [9] J. Hoffmann - Jοrgensen, Probability in Banach spaces, in: Ecole d'Eté de Probabilités de Saint-Flour VI, Lecture Notes in Math. 598, Springer, 1977, 1-186. 
  10. [10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, l988. Zbl0734.60060
  11. [11] A. V. Skorohod [A. V. Skorokhod], Studies in the Theory of Random Processes, Addison-Wesley, 1965. 

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